We consider a stage-structure Rosenzweig-MacArthur model describing the predator-prey interaction. Here, the prey population is divided into two sub-populations namely immature prey and mature prey. We assume that predator only consumes immature prey, where the predation follows the Holling type Ⅱ functional response. We perform dynamical analysis including existence and uniqueness, the positivity and the boundedness of the solutions of the proposed model, as well as the existence and the local stability of equilibrium points. It is shown that the model has three equilibrium points. Our analysis shows that the predator extinction equilibrium exists if the intrinsic growth rate of immature prey is greater than the death rate of mature prey. Furthermore, if the predation rate is larger than the death rate of predator, then the coexistence equilibrium exists. It means that the predation process on the prey determines the growing effects of the predator population. Furthermore, we also show the existence of forward and Hopf bifurcations. The dynamics of our system are confirmed by our numerical simulations.
Citation: Lazarus Kalvein Beay, Agus Suryanto, Isnani Darti, Trisilowati. Hopf bifurcation and stability analysis of the Rosenzweig-MacArthur predator-prey model with stage-structure in prey[J]. Mathematical Biosciences and Engineering, 2020, 17(4): 4080-4097. doi: 10.3934/mbe.2020226
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Abstract
We consider a stage-structure Rosenzweig-MacArthur model describing the predator-prey interaction. Here, the prey population is divided into two sub-populations namely immature prey and mature prey. We assume that predator only consumes immature prey, where the predation follows the Holling type Ⅱ functional response. We perform dynamical analysis including existence and uniqueness, the positivity and the boundedness of the solutions of the proposed model, as well as the existence and the local stability of equilibrium points. It is shown that the model has three equilibrium points. Our analysis shows that the predator extinction equilibrium exists if the intrinsic growth rate of immature prey is greater than the death rate of mature prey. Furthermore, if the predation rate is larger than the death rate of predator, then the coexistence equilibrium exists. It means that the predation process on the prey determines the growing effects of the predator population. Furthermore, we also show the existence of forward and Hopf bifurcations. The dynamics of our system are confirmed by our numerical simulations.
1.
Introduction
Mathematically, there are some complex phenomena in nature that are appealing to be investigated. One example is the interaction between prey and its predator. The first predator-prey interaction model was developed independently by [1] and [2], and therefore it is known as the Lotka-Volterra predator-prey model. Since then, the model for the interaction between prey and its predator is continuously studied and developed by both mathematicians and biologists [3]. One of important features that determine the characteristics of predator-prey interaction is the functional response which explains how many prey are eaten by predators per unit time. For example, the famous Rosenzweig-MacArthur predator-prey model implements the Holling-Type Ⅱ functional response p(x)=axb+x, where a and b are positive parameters related to the predator's attack rate and the handling time, respectively [4,5]. The Rosenzweig-MacArthur model has attracted many scholars to study the effect of various factors on predator-prey interactions, for example the effect of refuge [6,7,8,9], harvesting [10], quiescence [11], disease transmission among predators [12] or among prey [13] and the mutant population [14].
The transition between immature and mature stages from an adaptive dynamics perspective has recently been evaluated in [15]. In the predator-prey interaction, the age or size of the prey certainly determines the level of predation. Even in many cases, predators are only able to attack when the prey is still immature. Indeed, small size or immature prey tend to be more easily captured and eaten by predators. Moreover, there are many factors that significantly influence the dynamics of the predator–prey models with stage–structure. For example, [16] and [17] have studied the harvesting policy in a stage–structure predator–prey model. In [18], a ratio-dependent prey–predator model with stage–structure was investigated, and the uniform persistence and impermanence of the model were carried out through the sufficient conditions. Other features which influence the interaction of predator-prey with stage-structure have also been studied in many references, such as maturation delay [19,20,21], periodic functional responses [22,23], prey refuge [24], combination of prey refuge and additional food for predator [25], anti-predator [26,27,28], as well as diffusive effects [29].
Recently, the authors [30] have studied a Rosenzweig-MacArthur predator–prey model with stage–structure in prey:
where x1≡x1(t) and x2≡x2(t) are respectively the densities of immature and mature prey population at time t, and y≡y(t) is the density of predator population at time t. The following assumptions are taken in deriving model (1.1) :
(H1) The immature prey do not produce offspring and their growth rate depends entirely on the reproduction by the mature prey, where the growth is assumed to be logistically with constant intrinsic rate r>0 and constant carrying capacity K>0.
(H2) The immature prey is more vulnerable so predators only consume immature prey. The predation mechanism is assumed to follow the Holling Type Ⅱ functional response where the maximum predation rate is v0>0 and the environmental protection for immature prey is denoted by n1>0.
(H3) The immature prey grow up and turn into mature prey with a conversion rate α>0.
(H4) The growth of mature prey population depends only on the conversion of immature prey into mature prey. The mature prey do not have a risk to be attacked by predator. The death rate of the mature prey population is δ0>0.
(H5) The predator consumes only the immature prey and the growth rate of predator population is proportional to the predation rate, where the conversion rate of consumed prey into predator births is c>0. The death rate of the predator population is δ1>0.
Notice that the growth of immature prey in model (1.1) is inhibited by the intraspecific competition between the mature and immature prey. In nature, the intraspecific competition also occur between mature prey. Such intraspecific competition will be incoporated in this paper and hence we propose the following stage-structure predator-prey model
System (1.2) describes a predator-prey interaction where the predator only consumes immature prey. One ecological example of such a predator-prey system is the interaction between Moluccan megapode (Eulipoa wallacei) and rats Rattus sp.Rattus sp. is reported to prey upon Eulipoa eggs and chicks, and it cannot attack adult Eulipoa, see [31]. Another example of predator-prey system which can be described by model (1.2) is Chinese fire-bellied newt, which is unable to feed on the mature Rana chensinensis, can only prey on its immature, see [32]. For the sake of convenience, we describe all variables and parameters in system (1.2) in Table 1.
Table 1.
Description of variables and parameters in system (1.3). N stands for the number of individual per unit area.
Variable/Parameter
Description
Units
x1
Density of immature prey
N
x2
Density of mature prey
N
y
Density of predator
N
r
Intrinsic growth rate of prey
time−1
K
Carrying capacity of prey
N
α
Conversion rate of immature prey into mature prey
time−1
v0
Predation rate
time−1
n1
Environment protection for prey
N
c
Conversion rate of consumed prey into predator birth
We remark that the intraspecific competition between the mature and immature prey is assumed to have the same strength as that between mature prey. To simplify our analysis, we reduce the number of parameters in system (1.2) by considering the following non-dimensional model
where τ=rt, u1=x1K, u2=x2K, u3=v0yrK, ω=n1K, α1=αr, α2=cv0r, β1=δ0r and β2=δ1r.
As far as we are aware, the dynamics of system (1.3) have not been studied. Hence, a Rosenzweig-MacArthur predator-prey model with stage-structure in prey is introduced in this paper and the dynamical properties of the model are investigated. The organization of this paper is as follows. In Section 2, we analyze the boundedness, existence and uniqueness of the solution of the system (1.3). In Section 3 we determine all possible equilibrium points including the conditions for their existence and stability properties. The existence of Hopf-bifurcation and numerical simulations to illustrate our analytical finding are presented in Section 4 and Section 5, respectively. The conclusion of our study is given in the last section.
2.
Basic properties of the model
In this section we present some basic properties of the model (1.3) which include the existence and uniqueness, as well as the positivity and the boundedness of solutions of model (1.3). These are necessary because we deal with a population-related model. Therefore solutions of the model (1.3) must be non-negative. Boundedness of solutions explains that there are natural limits to population growth because resources are limited.
2.1. Existence and uniqueness of solutions
The existence and uniqueness of solutions of system (1.3) will be investigated in the region [0,∞)×ΩM where
ΩM={(u1,u2,u3)T∈R3+:max{|u1|,|u2|,|u3|}≤M}
for sufficiently large M. The existence of M is assured by the boundedness of the solutions, which will be discussed later. Let U=(u1,u2,u3)T, ˉU=(ˉu1,ˉu2,ˉu3)T and consider a mapping
where L1=(2α1+M+(1+α2)Mω), L2=(1+β1+3M), L3=(β2+(1+α2)(ω+M)M), and L=max{L1,L2,L3}. Clearly that H(U) satisfies the Lipschitz condition with respect to U, and therefore system (1.3) with any positive initial condition u1(0)≥0,u2(0)≥0,u3(0)≥0 has a unique solution U(τ)=(u1(τ),u2(τ),u3(τ))T∈ΩM. Hence, we have the following theorem related to the existence and uniqueness of the system (1.3).
Theorem 2.1.The stage-structure predator-prey system (1.3) subject to any non-negative initial value (u1(0),u2(0),u3(0)) has a unique solution (u1(τ),u2(τ),u3(τ))T∈ΩM for all τ>0.
2.2. Positive invariance
To show the positive invariance of the system (1.3), we first write the system (1.3) and its initial conditions in matrix form as
dUdτ=H(U),
(2.4)
where U(τ)=(u1(τ),u2(τ),u3(τ))T,U(0)=(u1(0),u2(0),u3(0))T∈ΩM and H(U) defined as in (2.1). From the previous subsection it was shown that H(U) is locally Lipschitz. Furthermore, it is obvious to show that Hi(Ui)|ui=0≥0, for i=1,2,3. Then, according to Nagumo theorem, the solution U(τ) of system (2.4) with initial condition U(0)=U0∈ΩM remains in ΩM for any τ≥0. Hence, we have the following theorem.
Theorem 2.2.Every solution of (1.3) with initial conditions u1(0)≥0,u2(0)≥0,u3(0)≥0 which exists in ΩM, remains positive for all τ>0.
2.3. Boundedness of solutions
Theorem 2.3.All solutions of system (1.3) which start ∈ΩM are uniformly bounded.
Proof. We begin by assuming that u1(τ)+u2(τ)≥1 for all τ≥0. Using the first two equations in system (1.3), we have
ddτ(u1+u2)=u2(1−(u1+u2))−β1u2−u1u3ω+u1.
(2.5)
It is clear that ddτ(u1+u2)≤0, and thus u1+u2 is monotone decreasing. Let's denote
limτ→∞u1+u2=κ.
(2.6)
Based on Barbalat lemma, we can show that if κ>1 then
This contradiction leads to κ=1. Since u1+u2 is monotone decreasing function, we have that
limτ→∞sup(u1+u2)=1,
(2.8)
meaning that there exists τ0>0 such that for τ>τ0 we get u1+u2≤1+ϵ or u1≤1+ϵ−u2 for any ϵ≥0. Then, from the second equation in system (1.3) we get for any τ≥τ0 that
Hence u1 and u2 is bounded above. The latter inequality shows that there exists τ1 such that u1≤β1α1+β1+ϵ for all τ≥τ1. Now define u=u1+u3α2. Using the first and the third equations in system (1.3), we obtain
By taking ϵ=0 and denoting η=|β2−α1|β1α1+β1, we get the following inequality
dudτ≤−β2u+η,
(2.14)
from which we have
u(τ)≤ηβ2+(u(0)−ηβ2)exp(−β2τ).
(2.15)
Hence, u3 is also bounded above because
limτ→∞supu≤ηβ2.
(2.16)
Now we suppose that our assumption that u1+u2≥1 is violated, i.e. there exists τ2>0 such that we have for the first time u1(τ2)+u2(τ2)=1. Then, we easily have
Consequently, whenever once a solution with u1+u2 has entered the interval (0,1) then it remains bounded there for all τ>τ2, i.e.
u1(τ)+u2(τ)<1,∀τ>τ2.
(2.18)
From the second equation in system (1.3) we get
du2dτ=αu1−β1u2≤α1−(α1+β2)u2.
(2.19)
Using the previous argument, we can prove that
limτ→∞supu2≤α1α1+β1.
(2.20)
Similarly, we can also show that
limτ→∞supu1≤β1α1+β1.
(2.21)
Again, by defining u=u1+u3α2 and using the previous argument, we can easily show that u is bounded above and therefore u3 is also bounded.
At last, if u1(0)+u2(0)<1, then by using the previous arguments, we have that u1(τ)+u2(τ)<1 and u=u1+u3α2 is bounded for all τ>0. This completes the proof.
3.
Existence and stability analysis of equilibrium points
It is easy to show that system (1.3) has three non-negative equilibrium points as follows.
● The extinction equilibrium E0=(0,0,0), which there is no population in the habitat.
● The predator-free equilibrium E1=(ˆu1,ˆu2,0), which exists if
β1<1
(3.1)
where ˆu1=β1(1−β1)/(α1+β1), ˆu2=α1ˆu1/β1.
● The interior equilibrium E∗=(u∗1,u∗2,u∗3), i.e. all of species coexist, where
The interior or coexistence equilibrium point exists if
β1<1,α2>β2,andω<˜ω,
(3.2)
where ˜ω=β1(α2−β2)(1−β1)β2(α1+β1).
From the description above, it can be seen that if the conditions (3.2) are fulfilled then all three equilibrium points exist. The local stability of each equilibrium points of system (1.3) is shown in the following theorem.
Theorem 3.1.For system (1.3), we have the following stability properties of its equilibrium points:
(i) The equilibrium point E0 is locally asymptotically stable if β1>1.
(ii) The equilibrium point E1 is locally asymptotically stable if ω>˜ω.
(iii) The coexistence equilibrium point E∗ is locally asymptotically stable if φ1>0,φ3>0 and φ1φ2>φ3 where φ1,φ2 and φ3 are defined as in the proof.
Proof. The local stability of all equilibrium points can be studied from the linearization of the system (1.3). The Jacobian matrix of the system (1.3) at a point (u1,u2,u2) is given by
By observing the eigenvalues of the Jacobian matrix (3.3) at each equilibrium point, we have the following stability properties.
(ⅰ) The Jacobian matrix of the system (1.3) at E0 has eigenvalues λ1=−β2 and λ2,3=−12B1±12√B21−4C1, where B1=α1+β1>0 and C1=α1(β1−1). If
β1>1
(3.4)
then C1>0. Consequently Re(λ2,3)<0 and E0 is locally asymptotically stable.
(ⅱ) The Jacobian matrix of the system (1.3) at E1 has eigenvalues λ1=β1(α2−β2)(1−β1)−β2ω(α1+β1)ω(α1+β1)+β1(1−β1) and λ2,3=−12B2±12√B22−4C2, where B2=α1+β1+α1u1/β1>0 and C2=α1(1−β1). If E1 exists then C2>0 and therefore Re(λ2,3)<0. Thus, E1 is locally asymptotically stable if λ1<0, that is when
ω>˜ω.
(3.5)
Notice that if E1 exists and is locally asymptotically stable, then E0 is unstable and E∗ does not exist.
(ⅲ) The characteristic equation of the Jacobian matrix of the system (1.3) at E∗ is given by the following cubic equation
From conditions (3.1) and (3.4), we note that if the intrinsic growth rate of immature prey is less than the death rate of mature prey then the extinction point E0 will be locally asymptotically stable. Otherwise, the predator-free point E1 exists whenever the death rate of mature prey is less than the intrinsic growth rate of immature prey. Thus, the existence of predator-free point E1 and the stability of the extinction point E0 are dependent on both intrinsic growth rate of immature prey and the death rate of mature prey. Next, condition (3.2) says that the coexistence equilibrium E∗ exists if the death rate of the predator is less than the predation rate.
Based on the existence and stability results of equilibrium points of the system (1.3), we remark that when ω>˜ω then the coexistence equilibrium point (E∗) does not exist. In this case, parameter β1 is a threshold: if it is greater than unity then the asymptotically stable extinction equilibrium point (E0) is the unique equilibrium point; if it is less than unity then E0 becomes unstable and there appears an asymptotically stable equilibrium point (E1). Hence the system (1.3) undergoes a forward bifurcation at β1=1. In fact a forward bifurcation may also be observed when β1<1, where in this case ω behaves as the threshold parameter. Since β1<1, the extinction point (E0) is unstable. Furthermore, when ω>˜ω, equilibrium point E1 is asymptotically stable and E∗ does not appear. On the contrary, if ω<˜ω then E1 becomes unstable and E∗ exists if α2>β2. If the Routh-Hurwitz criterion for the characteristics equation (3.6) is fulfilled then E∗ is asymptotically stable. Consequently, we have the following corollary about the occurrence of bifurcation in the system (1.3).
Corollary 3.2. (ⅰ) If ω>˜ω then the system (1.3) has a forward bifurcation from E0 to E1 around β=1.
(ⅱ) If β1<1, α2>β2, φ1>0,φ3>0 and φ1φ2>φ3 then the system (1.3) undergoes a forward bifurcation at ω=˜ω from E1 to E∗.
4.
Existence of Hopf-bifurcation
In this section, we study the Hopf-bifurcation around the coexistence equilibrium point E∗=(u∗1,u∗2,u∗3) of the system (1.3). We consider ω and α2 as the bifurcation parameters. ω=n1/K and α2=cv0/r are chosen as the bifurcation parameters because r and K are strongly related to the growth of immature prey, which controls energy input in the predator-prey system. Furthermore a,v0 and n1 are important parameters governing the exchange of energy from prey to predator.
Theorem 4.1.The stage-structure predator-prey system (1.3) undergoes Hopf-bifurcation around the coexistence equilibrium E∗ when ω passes through ω∗ where ω∗ satisfies φ(ω∗)=φ1(ω∗)φ2(ω∗)−φ3(ω∗)=0, φi(ω∗)>0 for i=1,2 and φ1(ω∗)φ′2(ω∗)+φ2(ω∗)φ′1(ω∗)−φ′3(ω∗)≠0.
Proof. At ω=ω∗, by the condition φ(ω∗)=0, the characteristic equation (3.6) can be written as
(λ2+φ2)(λ+φ1)=0.
(4.1)
If φ1=m1+m2−m3>0 and φ2=m1m2+m4+m5−m3m6>0, then the roots of the equation (4.1) are λ1=−φ1<0 and λ2,3=i√φ2. For any ω, the characteristic roots can generally be written as λ1(ω)=−φ1(ω), and λ2,3=μ1(ω)±iσ1(ω). By substituting λ(ω)=μ1(ω)+iσ1(ω) into equation (4.1), we have
Thus, if φ1(ω∗)φ′2(ω∗)+φ2(ω∗)φ′1(ω∗)−φ′3(ω∗)≠0, then transvesality condition is satisfied, and Hopf-bifurcation occurs when ω passes through ω=ω∗.
According to Theorem 4.1, there exists a Hopf bifurcation in the stage-structure predator-prey model (1.3) where the Hopf bifurcation is controlled by ω. In fact, using the same argument as in the proof of Theorem 4.1, we can show that the Hopf bifurcation can also be controlled by parameter α2. The possibility of the Hopf bifurcation occurance is stated in the following theorem.
Theorem 4.2.The stage-structure predator-prey system (1.3) undergoes Hopf-bifurcation around the coexistence equilibrium E∗ when α2 passes through α∗2 where α∗2 satisfies φ(α∗2)=φ1(α∗2)φ2(α∗2)−φ3(α∗2)=0 provided that φi(α∗2)>0 for i=1,2 and φ1(α∗2)φ′2(α∗2)+φ2(α∗2)φ′1(α∗2)−φ′3(α∗2)≠0.
5.
Numerical simulations
For graphical confirmation of the previous analytical results, several numerical simulations have been carried out. The model (1.3) is solved using the fourth-order Runge-Kutta method with some different initial conditions and some hypothetical values of the parameters. We first consider the following parameter values: α1=0.5,α2=β2=0.3, ω=0.03 and β1=1.01. Because β1>1, Theorem 3.1 says that the extinction of population point (E0) is the only equilibrium point which is locally asymptotically stable. This can be understood from the fact that in this case β1=δ0r>1, i.e. the death rate of mature prey is larger than the intrinsic growth rate of immature prey. Hence the system will asymptotically be convergent to the extinction equilibrium point. The behavior of this case is depicted in Figure 1.(a). Next, we carry out a simulation using the same previous parameter values, except β1=0.3. In addition to the unstable extinction of population point (E0), system (1.3) also has the predator-free equilibrium E1=(0.2625,0.4375,0.0). Since α2=β2, we have ˜ω=0 and therefore ω>˜ω. Based on Theorem 3.1, E1 is asymptotically stable. The case of α2=β2 is equivalent to the case of cv0=δ1. Hence, from the third equation in the system (1.2), we have that the per capita growth rate of predator is less than the predator death rate. This explains the extinction of predator population. The numerical solutions which describe this situation is plotted in Figure 1.(b). To see a complete view of the dynamics of the system (1.3) with parameter values α1=0.5,α2=β2=0.3, and ω=0.03, in Figure 2 we plot the equilibrium densities of immature and mature prey as function of β1. The equilibrium density of predator in this case is zero and thus it is not shown in the picture. Figure 1 shows that if β1<1 then E0 is unstable while E1 is stable. On the other hand, when β1>1, E1 disappears and E0 becomes stable. Thus, the system (1.3) undergoes a forward bifurcation.
Figure 1.
(Color figure online) Phase-portraits of the system (1.3) with parameter values: α1=0.5,α2=0.3, β2=0.3, ω=0.03, and (a) β1=1.01, (b) β1=0.3. The red and green circles represent unstable and stable equilibrium point, respectively.
Figure 2.
(Color figure online) The forward bifurcation diagram from E0 to E1 for system (1.3). The red and blue lines represent the equilibrium point E0 and E1, respectively. The dashed-line indicates unstable equilibrium while the solid-line indicates stable equilibrium.
We now choose α1=α2=0.5 and β1=β2=0.3. Using Theorem 4.1, we can check that the system undergoes a Hopf bifurcation around E∗ where the bifurcation point is at ω∗=0.03989. To verifiy the existence of Hopf bifurcation numerically, we take two different values of ω, i.e. ω=0.03<ω∗ and ω=0.04>ω∗. For ω=0.03, we find that the system (1.3) has coexistence equilibrium point E∗=(0.045,0.075,0.072), with φ1=1.2617>0 and φ3=0.0338>0, but φ1φ2−φ3=−0.0323<0. Since the Routh-Hurwitz criterion is not satisfied, the coexistence equilibrium point is unstable. However, for ω=0.04, the coexistence equilibrium point E∗=(0.06,0.1,0.09) is stable because the Routh-Hurwitz criterion is satisfied, i.e. φ1=1.26>0, φ3=0.0324>0 and φ1φ2−φ3=0.00036>0. The numerical solutions of the system (1.3) for ω=0.03 and ω=0.04 are shown in Figure 3. It can be seen in this figure that our numerical solutions are convergent to a periodic solution when ω=0.03<ω∗ and convergent to E∗ when ω=0.04>ω∗. This behavior indicates that the coexistence equilibrium point (E∗) changes its stability, i.e. E∗ is unstable and convergent to a limit cycle when ω<ω∗ and it becomes asymptotically stable if ω>ω∗. In other words, the system (1.3) experiences a Hopf bifurcation driven by parameter ω. Figure 3 gives an indication that the Hopf bifurcation is supercritical because the limit cycle is stable,
Figure 3.
(Color figure online) Phase-portraits of the system (1.3) where α1=α2=0.5,β1=β2=0.3, and (a) ω=0.03, (b) ω=0.04. The red and green circles represent unstable and stable equilibrium point, respectively. A solution with an initial value, which is close to equilibrium point E0, initially approaches equilibrium point E1 but then continues to move towards the stable limit cycle (a) or equilibrium point E∗ (b).
As stated in Theorem 4.2, the Hopf bifurcation can also be controlled by parameter α2. To show this behavior we perform simulation using α1=0.5,β1=β2=0.3 and ω=0.03. The critical α∗2 for the occurance of Hopf bifurcation can be determined using Theorem 4.2, where in this case we get α∗2=0.5822. Using Theorem 3.1, it can be verified that the coexistence equilibrium point (E∗) is unstable for the case of α=0.55, and it is asymptotically stable for α=0.59. As depicted in Figure 4, such stability properties coincide with our numerical simulations. It is clearly seen in Figure 4.(a) that for α2=0.55<α∗2, E∗ is unstable and the solutions converge to a limit cycle (periodic solution). If we take α2=0.59>α∗2, then the solution is convergent to E∗, see Figure 4.(b). From Figure 4, we also see the occurance of (supercritical) Hopf bifurcation controlled by α2.
Figure 4.
(Color figure online) Phase-portraits of the system (1.3) where α1=0.5,β1=β2=0.3,ω=0.03, and (a) α2=0.55, (b) α2=0.59. The red and green circles represent unstable and stable equilibrium point, respectively. A solution with an initial value, which is close to equilibrium point E0, initially approaches equilibrium point E1 but then continues to move towards the stable limit cycle (a) or equilibrium point E∗ (b).
To see the detail dynamics of the system (1.3) with parameter α1=0.5,β1=0.3 and β2=0.3, we plot the stability region in (ω,α2)−plane, see Figure 5. In this picture we see that there are three different regions: the cyan region represents the stability area of E1 (E0 exists but it is unstable, E∗ does not exist); the green region represents the stability area of E∗ (E0 and E1 exists but they are not stable); and the yellow region corresponds to the stable limit cycle (all the three equilibrium points exists but they are unstable). We identify the occurrence of a Hopf bifurcation as a change from the green region to the yellow region due to the changes of parameters ω or α2. We also notice the occurrence of a forward bifurcation caused by the changes of α2, see the changes from cyan region to green region. To provide a clearer illustration, we plot a bifurcation diagram for system (1.3) with parameter α1=0.5,β1=β2=0.3 and ω=0.03, see Figure 6. From this figure we can see the appearance of transcritical bifurcation which is caused by the changing of the value of α2. When α2<α12=0.3343, the equilibrium point E1 is asymptotically stable. On the contrary, if α12<α2<α22=0.3922 then equilibrium point E1 becomes ustable and equilibrium point E∗ is asymtotically stable. In Figure 6 we also observe that the system (1.3) undergoes 2-Hopf bifurcation. In this case, the equilibrium point E∗ is asymptotically stable for α12<α2<α22=0.3922 or α2>α∗2=0.5822, and it is unstable for α22<α2<α∗2. In the latter case, the system (1.3) is convergent to a periodic solution or limit cycle.
Figure 5.
(Color figure online) Bifurcation diagram in (ω,α2)-plane for the system (1.3) with parameter α1=0.5,β1=0.3 and β2=0.3. The black circles correspond to the parameter values used in Figure 1.(b), Figure 3 and Figure 4, respectively.
Figure 6.
(Color figure online) Transcritical and Hopf bifurcation diagram for system (1.3) with parameter α1=0.5,β1=β2=0.3 and ω=0.03. The black and blue lines represent the stable equilibrium point E1 and E∗, respectively. The red lines indicates the stable limit cycle.
A Rosenzweig-MacArthur predator-prey model with stage-structure in prey has been discussed. It was shown that the proposed model has three equilibrium points, i.e. the extinction of population (E0), the predator-free point (E1), and the coexistence point (E∗). All of these equilibrium points are conditionally asymptotically stable. Our analysis also showed that the proposed model exhibits a Hopf-bifurcation which can be driven by (ω) or (α2). Furthermore, the proposed model may also undergo a forward bifurcation for suitable parameter values. The analytical findings have been confirmed by our numerical simulations.
Acknowledgments
This research was funded by FMIPA via PNBP-University of Brawijaya according to DIPA-UB No. DIPA-023.17.2.677512/2020, under contract No. 12/UN10.F09/PN/2020.
Conflict of interest
All authors declare that there are no conflicts of interest regarding the publication of this paper.
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Conversion rate of consumed prey into predator birth
dimensionless
δ0
Death rate of mature prey
time−1
δ1
Death rate of predator
time−1
Figure 1. (Color figure online) Phase-portraits of the system (1.3) with parameter values: α1=0.5,α2=0.3, β2=0.3, ω=0.03, and (a) β1=1.01, (b) β1=0.3. The red and green circles represent unstable and stable equilibrium point, respectively
Figure 2. (Color figure online) The forward bifurcation diagram from E0 to E1 for system (1.3). The red and blue lines represent the equilibrium point E0 and E1, respectively. The dashed-line indicates unstable equilibrium while the solid-line indicates stable equilibrium
Figure 3. (Color figure online) Phase-portraits of the system (1.3) where α1=α2=0.5,β1=β2=0.3, and (a) ω=0.03, (b) ω=0.04. The red and green circles represent unstable and stable equilibrium point, respectively. A solution with an initial value, which is close to equilibrium point E0, initially approaches equilibrium point E1 but then continues to move towards the stable limit cycle (a) or equilibrium point E∗ (b)
Figure 4. (Color figure online) Phase-portraits of the system (1.3) where α1=0.5,β1=β2=0.3,ω=0.03, and (a) α2=0.55, (b) α2=0.59. The red and green circles represent unstable and stable equilibrium point, respectively. A solution with an initial value, which is close to equilibrium point E0, initially approaches equilibrium point E1 but then continues to move towards the stable limit cycle (a) or equilibrium point E∗ (b)
Figure 5. (Color figure online) Bifurcation diagram in (ω,α2)-plane for the system (1.3) with parameter α1=0.5,β1=0.3 and β2=0.3. The black circles correspond to the parameter values used in Figure 1.(b), Figure 3 and Figure 4, respectively
Figure 6. (Color figure online) Transcritical and Hopf bifurcation diagram for system (1.3) with parameter α1=0.5,β1=β2=0.3 and ω=0.03. The black and blue lines represent the stable equilibrium point E1 and E∗, respectively. The red lines indicates the stable limit cycle