Stability of a stage-structure Rosenzweig-MacArthur model incorporating Holling type-II functional response

The local stability of the Rosenzweig-MacArthur predator-prey system with Holling type-II functional response and stage-structure for prey is studied in this paper. It is shown that the model has three equilibrium points. The trivial equilibrium point is always unstable while two other equilibrium points, i.e., the predator extinction point and the coexistence point, are conditionally stable. When the predation process on prey increases, the number of predator increases. If the predation rate is less than or equal to the reduction rate of the predator, then the predator will go to extinct. By using the Routh-Hurwitz criterion, the local stability of the interior equilibrium point is investigated. It is also shown that the model undergoes a Hopf-bifurcation around the coexisting equilibrium point. The dynamics of the system are confirmed by some numerical simulations.


Introduction
Continuously, various studies on predator-prey interactions in the mathematical ecology are carried out [1]. There are several models known for studying the behavior of predator-prey systems, one of which is the Rosenzweig-MacArthur model [2]. The fundamental aspect about the Rosenzweig-MacArthur model is a well-known mechanism of the paradox of enrichment [3][4][5][6]. This is related to the extinction of species affected by the increasing of carrying capacity of prey in the ecosystem [4][5]. The effects of refuge on the Rosenzweig-MacArthur model have been studied by Kar [7] and Chen et al. [8]. Recently, studies of the dynamic behavior of fractional order Rosenzweig-MacArthur system with harvest and prey refuge have been studied by Javidi and Nyamoradi [9] and Moustofa et al. [5], respectively.
In studying predator-prey systems, functional responses are one of the important elements that affecting this dynamic, and one of them is the Holling-II type functional response [1,10]. Stage-structure is also considered to be one of the important factors in the predator-prey system. In some species, it is divisible into two stages, immature and mature [1,[10][11][12][13][14][15][16][17].
In addition to the stage structure on the predator [11][12][13][14][15], various studies have been carried out with a consideration of the stage structure on the prey. Khajanchi and Banerjee [16] analysed the role of prey refuge of the predator-prey model incorporating ratio-dependent functional response and considered stage-structure for the prey. The model analysis has shown that the stage-structured predator-prey species is significantly influenced by the prey refuge. Falconi et al. [17] considered a group defense mechanism and stage-structure for the prey. The analyzed model, assumed that immature and mature prey do not inhabit in the same location. It is found that the defense mechanism of the prey population has an impact the existence of the immature and mature prey and the predator. Next, stability analysis of a stage-structure model incorporating Holling type-IV functional response and considered competition on the mature prey was studied by Jia and Wei [1].
Motivated by these various studies, we consider the following the Rosenzweig-MacArthur system incorporating stage-structure for prey and Holling type-II functional response. Here, we consider a mathematical model of the interaction between the densities of immature prey 1 , mature prey 2 , and predator . By assuming that the predator just preys upon the immature prey, we get the following model Here, and are respectively the intrinsic growth rate of immature prey and carrying capacity of immature prey. and 0 are the transition rate from the immature prey to the mature prey and the maximum predation rate. 1 and are the environment protection to immature prey and the predator conversion rate. The death rate of mature prey and the predator are respectively 0 and 1 . All parameters , , , 0 , 1 , and in the system (1) are positive.
We outline the format of this paper as follows. In section 2, we analyze our mathematical model which includes the determination of equilibrium points, local stability analysis and Hopf-bifurcation analysis of the coexistence point. Some numerical simulations are performed to corroborate our analytical finding in section 3. In the last section 4, we conclude the results of our analysis accompanied numerical simulations.

Equilibrium points analysis
By setting the right-hand sides of the system (2) equal to zero, we get the following equilibrium points: (1) A trivial equilibrium 0 = (0,0,0) which is the extinction of all population in the ecosystem.
(2) The predator extinction equilibrium 1 = (̂1,̂2, 0), which exists if Notice that the predator extinction equilibrium 1 exists if the intrinsic growth rate of immature prey (r) greater than death rate of mature prey ( 0 ).
, and 3 * =  (4) says that if the predation rate is larger than the death rate of the predator, then the coexistence equilibrium * exists. It means that the predation process on prey determines the growing effects of the predator population.

Local stability of equilibrium point
The local stability of all equilibrium points can be studied from the linearization of system (2). The Jacobian matrix of the system (2)  .
By observing the eigenvalues of the Jacobian matrix (6) at each equilibrium point, we have the following stability properties.
Thus, the transversality condition be in force, and Hopf-bifurcation come to pass at 2 = 2 * .

Numerical Simulations
From the previous analytical proceeds, we present some numerical results. In consequence of all parameters values of the model are not available, we use hypothetical parameters: α 1 = 0.91, α 2 = 0.09, = 0.08, 1 = 0.06, 2 = 0.09. Using these values, we have ( 2 − 2 )̂1 − 2 = −0.0072 < 0, and hence the predator extinction point 1 = (0.940,1.426,0) is asymptotically stable and * does not exist. This means that immature and mature prey will survive in the system, while predator will go extinct. This situation is clearly shown by our numerical result shown in Figure. 1. If we increase the value of α 2 , i.e. using α 2 = 0.12, then we have that ( 2 − 2 )̂1 − 2 = 0.0228 > 0, 1 = 0.780 > 0, 2 = 0.002 < 0, and 4 = −0.003 < 0. Observe that the stability conditions for both extinction of predator point and coexistence point are not satisfied and therefore 1 and * are unstable. This situation is shown in Figure 2. Indeed, the numerical solution is not convergent to any equilibrium point, but there appears a periodic solution. If we further increase the parameter value of α 2 such that α 2 = 0.14 and consistently using the same parameter, we obtain ( 2 − 2 )̂1 − 2 = 0.042 > 0, 1 = 0.800 > 0, 2 = 0.005 > 0, and 4 = 0.001 > 0. Hence, the coexistence equilibrium point * = (1.440,0.214,0.270) is stable. This behavior is confirmed by our numerical simulation as depicted in see Figure 3. From Figure 2 and Figure 3, we see that there occurs a Hopf bifurcation which is driven by α 2 .

Conclusions
From the analysis of the system (2), we obtain three equilibrium points namely the extinction populations point 0 , the extinction of predator point 1 , and the coexistence point * . 0 is always unstable, whereas 1 and * are stable under certain conditions. If ( 2 − 2 )̂1 < 2 then 1 stable and * does not exist. Increasing the parameter value of the predation rate may stabilize equilibrium * . Here, Hopf-bifurcation around coexistence equilibrium point * with respect to the predation rate. The analysis shows that if 2 is less than or equal to 2 then it may lead to disappearance of coexistence equilibrium. Otherwise, if 2 is greater than 2 then the coexistence equilibrium point exists and is stable.