Bifurcation Analysis on Predator-Prey Leslie-Gower Model Holling Respond Function Type II With Harvesting on Predator and Prey Population

L Setyawan1, Hartono2
Department of Mathematics Education, Yogyakarta State University
lilik.setyawan2016@student.uny.ac.id1, hartono@uny.ac.id2

Abstract. The objective of this research is to study Leslie-Gower predator-prey Holling respond function type II models on predator and prey population. In this predator-prey model it is assumed that there are harvesting efforts in both populations.The steps to analyze this models including to looking for equilibrium points, and analyzing the exchange  the parameter of permanent harvesting value to know the possibility of bifurcation. Based on analysis obtained four equilibrium points which is the trivial equilibrium point , the equilibrium point  for the extinction of  predator, the equilibrium point  for the extinction of prey, and the endemic equilibrium point . The equilibrium points of  and  experiencing stability changes when the rate of  harvesting parameter on prey population () and predator () is varied so that the system have a bifurcation. The forming of limit cycle on photo phase around equilibrium points shows that the kind of bifurcation that happen was a Hopf bifurcation.
1.	Introduction 
Ecology is the study of the reciprocal relationship between species and their environment and non-living objects [1]. Ecology can also be defined as the study of both the interaction between living things and the interaction between living things and their environment. There is a mutually beneficial interaction between living things, which is known as symbiotic mutualism. If it is not mutually beneficial for the two species, then the interaction is called competition interaction. Meanwhile, if one of the species experiences a loss, the interaction that occurs is the prey-predator interaction.
The interaction between two populations that prey on each other is usually referred to as predator-prey. The predator-prey model was first introduced by Lotka in 1925 and Volterra in 1926. Lotka-Volterra research resulted in a simple model of predation or interaction between two species in an ecosystem and subsequently also introduced the classic Lotka-Volterra model which has recently been developed by researchers. In 1934 Gause failed to describe the growth rate of the two populations he obtained with the Lotka-Volterra model, then Leslie (1948) was able to obtain a model representing the growth data obtained by Gause [2].
In 1948, Leslie and Gower introduced a model with a predatory population growing following a logistical model so that its growth was limited [3]. In addition, the predatory growth rate is proportional to the number of prey populations. Thus, if we denote x (t) as the prey population density and y (t) the predatory population density, the Leslie-Gower model is given to the following system of differential equations:

	(1.1)	
In this model the predation response function is added. The predation response function is the level of predation (feeding power) of the predator on the number of prey. The predation response function used in this model is the Holling type II response function. The choice of the Holling type II response function conforms to the type of predator that is active and takes time to digest prey, as did the wolf [4]. Furthermore, in this study it was assumed that both populations experienced harvesting by humans.
2.	Discussion
2.1	Model mathematics
Model formation is based on the assumptions made. Assumptions are used so that the mathematical model obtained can present the actual situation. The assumptions used are as follows:
a.	The Predator-Prey Leslie-Gower model consists of two populations, namely the predator population and the prey population.
b.	The interaction between a predatory population and a prey population follows the Holling type II response function.
c.	The number of predatory population growth follows the logistical growth pattern.
d.	In both populations a harvesting effort was carried out.
e.	The birth rates of the prey and predator populations are greater than the harvest rates of the two populations.
Based on the above assumptions, the Prey Leslie-Gower predator model, the Holling type II response function with harvesting, is as follows:
	(2.1)	
		
The definitions and variable names used in the above model are presented in the table below
Table 1. Definition of Variables and Parameters
Variables and Parameters	Definition	Terms	
	Times		
	Prey population		
	Predator population		
	The current prey population t		
	The current predator population t		
	Prey population birth rate		
	Predator population birth rate		
	Prey population mortality rate due to competition between individuals		
	Prey population mortality rate due to interactions with predatory populations		
	Predator population mortality rate due to interactions with predatory populations		
	The level of protection of the prey population from predatory populations by the environment		
	Level of protection against predators		
	Prey population harvest rate		
	Predator population harvest rate		
2.2	System with two independent parameters
In System (2.1), two parameters will be selected to be independent, namely  and , while the other parameters are fixed. Fixed parameter values ​​are presented in the following table:

Table 2. Parameter Values
Parameter	Value	
	12	
	0.01	
	7.5	
	4	
	3	
	4	
	6	

The parameter values ​​in the table above are then substituted into the system (2.1), except for the prey population harvest rate parameter () and the harvest rate parameter for the predator population (), so that the following new system is obtained,
	(2.2)	

2.2.1	Equilibrium points
The equilibrium points of the above system are obtained by satisfying   and  [5]. Suppose  is the equilibrium point of System (2.2) so that the equilibrium points are obtained as follows
a.	
b.	
c.	
d.	

2.2.2	Stability analysis of the equilibrium points
The stability of the equilibrium point in System (2.2) can be checked using the Jacobian matrix  [6]. The Jacobian Matrix of System (2.2) is as follows,
	(2.3)	

a.	Stability analysis at equilibrium point 
To analyze the stability of the system (2.2) at the equilibrium point (0.0), substitute the  point value which corresponds to equation (2.3) so that the characteristic equation is obtained

so that the eigenvalues ​​are obtained as follows:

Because the eigenvalue > 0 and > 0, the stability at point (0,0) is unstable.
b.	Stability analysis at equilibrium point 
To analyze the stability of the system (2.2) at the equilibrium point , substitute the  point value which corresponds to equation (2.3) so that the characteristic equation is obtained

so that the eigenvalues ​​are obtained as follows:

Because the eigenvalues ​​< 0 and > 0 then the stability at point  is unstable.
c.	Stability analysis at equilibrium point 
To analyze the stability of the system (2.2) at the equilibrium point , substitute the  point value that corresponds to equation (2.3) so that the characteristic equation is obtained

so that the eigenvalues ​​are obtained as follows:

The value of  depends on the parameters  and , if = 3 and = 3 will be obtained = 0 so that bifurcation is possible.
d.	Stability analysis at equilibrium point 
To analyze the stability of the System (2.2) at the equilibrium point , substitute the parameters in Table 2 to System (2.3) so that the following system is obtained

Furthermore, we will look for the characteristic equation of the system above so

where  and . The equilibrium point  is stable if it satisfies   dan 

2.3	Simulation of a system with two independent parameters
Case 
In this case,  and  are chosen so that the system (2.2) becomes:
	(2.4)	
		
In the system above, there are five equilibrium points, namely ,, ,  dan .The phase portrait of the System (2.4) using Maple 18 is as follows,
	
Figure 1. Phase portrait  and 	
In Figure 1, it can be seen that the trajectory will know point  in the direction of the x axis and the y axis so that point  is unstable, the trajectory approaches point  in the direction of the x axis then moves away from the direction of the y axis so point  is unstable, at point  the trajectory is approaching the direction of the y axis then moving away from the direction of the x axis so that point  is unstable, the trajectory approaches point  in the direction of the y-axis and its surroundings so that point  is stable, then at point  all trajectories approach from various directions so that point  is stable. So it can be seen that the equilibrium points ,  and  are unstable while the equilibrium points  and  are stable. The stability of the equilibrium point can also be determined based on the eigenvalues ​​of the linearization results around the equilibrium point. From the system (2.4), the Jacobian Matrix is ​​obtained as follows,

Based on the Jacobian Matrix above, for point  two positive eigenvalues ​​are obtained so that point  is the source, for points  and  two eigenvalues ​​are obtained positive real and negative real so that points  and  are saddle, for point  two eigenvalues ​​are negative real so that point  is a sink, while for point , two eigenvalues ​​are obtained in the form of a complex number with a negative real part so that point  is a spiral.
Case 
For  and , System (2.2) becomes:
	(2.5)	
		
In the above system, there are five equilibrium points, namely ,, ,  and . The phase portrait of the System (2.5) using Maple 18 is as follows,
	
Figure 2. Phase portrait  and 	
In Figure 2, it can be seen that the trajectory will know point  in the direction of the x-axis and the y-axis so that point  is unstable, the trajectory approaches point  on the x-axis direction then moves away from the y-axis so point  is unstable, at point  the trajectory is approaching the direction of the y axis then moving away from the direction of the x axis so that point  is unstable, the same thing happened at point  all trajectories move away from various directions so that point  is unstable, then at point  all trajectories approach from various directions so that point  is stable. So it can be seen that the equilibrium points ,,  and  are unstable while the equilibrium point  is stable. The stability of the equilibrium point can also be determined based on the eigenvalues ​​of the linearization results around the equilibrium point. From the system (2.5), the Jacobian Matrix is ​​obtained as follows,

Based on the Jacobian Matrix above, for point  two positive eigenvalues ​​are obtained so that point  is the source, for points  and  two eigenvalues ​​are obtained positive real and negative real so that points  and  are saddle, for point  two eigenvalues ​​are negative real so that point  is a sink, while for point , two eigenvalues ​​are obtained in the form of a complex number with a positive real part so that point  is a spiral.
For  dan , System (2.2) becomes:
	(2.6)	
		
In the system above, three equilibrium points are obtained, namely ,, . The phase portrait of the System (2.6) using Maple 18 is as follows,
	
Figure 3. Phase portrait  and 	
In Figure 3, it can be seen that the trajectory will know point  in the direction of the x axis and the y axis so that point  is unstable, the trajectory approaches point  on the x-axis direction then moves away from the y-axis so that point  is unstable, then at point  all trajectories approach from various directions so that point  is stable. So it can be seen that the equilibrium point , dan  is unstable while the equilibrium point  is stable. The stability of the equilibrium point can also be determined based on the eigenvalues ​​of the linearization results around the equilibrium point. From the system (2.6), the Jacobian Matrix is ​​obtained as follows,

Based on the Jacobian Matrix above, for point  two positive eigenvalues ​​are obtained so that point  is the source, for point  two eigenvalues ​​are real positive and negative real so that point  is a saddle, for point  it is obtained two negative real values ​​of eigenvalues ​​so that point  is a sink. 
Based on the phase portrait in the images above, a diagram of equilibrium points and the stability of the equilibrium point of the System (2.2) is compiled as follows:
	
Figure 4. Bifurcation diagram	
Changes in the stability of the equilibrium point due to changes in parameter values ​​are called bifurcations. In Figur 4, it can be seen that there is a change in the stability of the  and  equilibrium points from the system (2.2) due to changes in the value of the parameters  and . At the equilibrium point  is stable when the parameter value  while it is unstable when the parameter value . Then at the equilibrium point  is stable when the parameter value , while when the parameter value  experiences instability. The change in stability at the equilibrium points  and  indicates that System (2.2) is bifurcated [7]. The appearance of the limit cycle in the phase portrait that occurs in System (2.2) around the  equilibrium point and the change in many equilibrium points and changes in the stability of the equilibrium point indicate that the bifurcation that occurs in System (2.2) is a Hopf bifurcation [8].

3.	Conclusion
Predator-prey Leslie-Gower model Holling respond function type II with harvesting on predator and prey population having four equilibrium points, namely  and . The equilibrium points  and  experience a change in stability when the value of the prey () and predator () population harvesting parameters is varied while the parameter values ​​  and  are fixed (fixed) so that the system is bifurcated. The formation of the limit cycle in the phase portrait of the system around the equilibrium point  shows that the type of bifurcation that occurs in the Leslie-Gower predator-prey mathematical model with the Holling type II response function harvesting in the predator and prey population is Hopf bifurcation.

4.	References 

[1]	Suzyanna 2013 Interaction between Predator-Prey and Prey Harvesting Factors Journal of Scientific Modeling & Computation 1 60
[2]	Berryman A A 1992 The Origins and Evolution of Predator-Prey Theory. Ecology. Amerika 73 1531
[3]	Afiyah S N 2015 Dynamic Analysis of the Predator-Prey Leslie-Gower Model with the Holling Type II Response Function  ASIA Scientific Journal of Technology and Information 9 2
[4]	Ruan S and Xiao D 2001 Global Dynamics of a Ratio-Dependent Predator-Prey System Mathematical Biology 43 269
[5]	Perko L 2001 Differential Equations and Dynamical Systems. 3th Edition (New York: Springer)
[6]	Purnamasari D Faisal and Noor A J 2009 Predator-Prey Leslie System Stability Journal of Pure and Applied Mathematics 3 52
[7]	Kuznetsov Y A 1998 Elements of Applied Bifurcation Theory 2nd Edition (New York: Springer)
[8]	Lynch S 2010 Dynamical Systems with Application using Maple 2nd Edition (Berlin: Birkhauser)