Dynamic Analysis of a Prey Predator Model with Holling-Type III Functional Response and Anti-Predator Behavior

The prey-predator model in this article reviews the interaction of two populations with a type III Holling -type III functional response and anti-predator behavior. Dynamic analysis starts by determining the basic model construction assumptions, equilibrium and stability points, and numerical simulations using Python. Dynamic analysis results obtained four equilibrium points with types of stability, namely 𝑬 𝟎 (𝟎, 𝟎) which is unstable, 𝑬 𝟏 (𝒌, 𝟎) which is asymptotically stable 𝑬 𝟐 , and 𝑬 𝟑 . Which is stable under certain conditions. The numerical simulation results show double stability at the equilibrium points. 𝑬 𝟏 and 𝑬 𝟐 with the anti-predator behavior parameter value 𝜼 = 𝟎. 𝟎𝟏 . The anti-predator parameter value 𝜼 = 𝟎. 𝟒𝟕𝟑 indicates a change in stability that is only the 𝑬 𝟏 equilibrium point. Differences in the values of the anti-predator behavior parameters affect changes in system solutions and impact reducing predator populations.


Introduction
All populations on Earth cannot live alone but need and interact with each other.The interaction process includes the process of eating and being eaten by one animal species with another animal.Animals that eat other animals are called predators, while animals that other animals hunt are called prey [1].The purpose of interaction between predators and their prey is to control the density of prey populations to maintain their lives [2], [3].
Alfred Lotka and Vito Volterra introduced the prey-predator interaction model, namely the Lotka-Voltera model [4].In 1953, Holling introduced an expanded Lotka-Voltera model of predation called the response function Holling proposed four types of response functions: type I, type II, type III, and type IV response functions [5], [6].
In general, the interaction process between prey and predators is assumed to be that the prey group is always caught and eaten by predators [7], [8].Not all prey can be captured and eaten by predators, but some can attack or eat other individuals.In this case, some prey could fight and escape the predator.Such is the case with the Red Colobus Monkey (Piliocolobus), which protects itself by shouting alarm calls for its group when threatened by Chimpanzees [9].Therefore, pressure from predators that causes prey to escape and fight back from predators is known as anti-predator behavior [10].
Researchers developed the Lotka Volterra prey-predator model, which involves anti-predator behavior.Until now, few studies have discussed the dynamic analysis of the prey-predator system with the Type III Holling response function and anti-predator behavior.This research refers to two studies, including [11] studied a predator-prey model with a Holling Type IV response function and anti-predator behavior.Supported by other studies, [12]- [14] examines the interaction model of prey-predator with ratio-dependent function and antipredator behavior.The complex dynamics of the model with the phenomenon of anti-predator behavior in nature have also been discussed [15]- [17].
Inspired by these two studies, researchers still consider anti-predator behavior by modifying it using the Holling type III functional response.Holling type III active response is suitable because it has the problem of predators that tend to look for other prey populations when the prey population continues to decrease due to self-protection, namely anti-predator behavior.

Figure 1.Research Flow Chart
Based on Figure 1, the stages begin with a literature study, constructing a model related to assumptions, dynamic analysis, which includes equilibrium points, determining the type of stability, determining parameters for numerical simulations with a Python program illustrated with phase portraits, and drawing conclusions based on the results of the analysis.

Construction of a Prey-Predator Model
Based on the same assumption in [6], [11], the prey-predator interaction model obtained uses a different functional response, namely Holling type III and anti-predator behavior.The growth rate of the prey population follows the logistic equation.Construction is denoted. (1) The dynamical system (1) has been analysed in the region {(, ) ∈ ℝ + 2 :  ≥ 0,  ≥ 0} with the initial conditions (0) > 0, (0) > 0. Here, () dan () denote the numbers of prey and predator.The predators consume the prey with Holling type III functional response The equilibrium point in a model ( 1) is obtained by solving the system of equations that makes   = 0 dan   = 0 [18].System (1) has different points of equilibrium. 0 ,  1 , and  2 .
Let's say  =  * with  as the value of the equilibrium point  * which is obtained by simplifying the equation ( 1) so the value of  * must be positive as follows.
Here is the value of each component. .

b. Local Stability of Equilibrium
Stability analysis is determined by the linearization process on the system arranged in the form of a matrix [21].The linearization results use the following Jacobian matrix: We obtain the following theorem of the stability of the system's equilibrium (1).Theorem 1. Equilibrium point   (, ) is unstable (saddle point) Proof.The Jacobian matrix in equation ( 4) at  0 (0,0) is as follows: The eigenvalues Jacobian matrix ( 5) is.

c. Numerical Simulation
Numerical simulations show dynamic behavior changes around the equilibrium point [23].To perform a numerical simulation, the values of the following parameters are required.Based on parameters in Table 1, with  = 0.1 four equilibrium points exist from the eigenvalue results to know the type of stability.In Table 3, two equilibrium points are obtained that meet the conditions for the existence of stability, namely, the equilibrium point at  1 and equilibrium point  2 .Therefore, a system with stability at two equilibrium points is called bistable.Stability changes of system behavior in equations ( 1) and (2) are shown through numerical simulations by increasing the value of the anti-predator behavior parameter the first  = 0.1 with  = 0.473.The initial values used for the numerical simulations are the same as those [17,17] and [1,17], which produce phase portraits.Numerical simulation results based on the parameter values in Table 2 with different parameter values on anti-predator behavior  = 0.1 illustrated in Figure 2 and Figure 3.The system (1) can exhibit bistability at η = 0.1.In Figure (2) and Figure (3), the system has double stability, namely bistable at  1 and  2 influenced by differences in initial values.Figure (2) shows the solution from a convergent system to an equilibrium point  1 (7.5,0) using initial values  1 [17,17] so that the equilibrium point  1 is stable.Figure 3 shows the solution from a convergent system to an equilibrium point  3 (0.04, 4.66) using initial values  2 [1,17] so that the equilibrium point  2 is stable.7 show that parameter anti-predator behavior is  = 0.473 produce system has stability only at  1 with the difference in the initial value  1 [17,17] and  2 [1,17].Both show a solution from a convergent system to an equilibrium point. 1 (7.5,0) so that the equilibrium point  1 are stable.
The results of the two simulations show that anti-predator behavior can affect a change in the system's stability.When the parameter is anti-predator behavior  = 0.1 results in a bistable system that is at the equilibrium point  1 which states the population of predators experiencing extinction and the equilibrium point  2 states that both populations exist.The appearance of a bistable system indicates that the system solution is influenced by the initial values of the two people.When the parameter of anti-predator behavior increases to  = 0.473 it produces stability, namely at the equilibrium point  1 the predator population is experiencing extinction because they cannot live without predation.The results of the Dynamic Analysis produce three equilibrium points, namely. 0 (0,0) declaring extinction in two populations,  1 (, 0) declared prey populations to extinction, and  2 as well as  3 states that both populations exist.Stability analysis of the equilibrium points with the type of stability of each, namely  0 unstable.Except for  1 ,  2 , and  3 stable under certain conditions of existence.Differences in the parameter values of anti-predator behavior are carried out to determine the effect of anti-predator behavior on changes in system solutions.The numerical simulation results show a bistable system when the parameter anti-predator behaviour is  = 0.1, which means that both prey populations can coexist and the predator populations experience extinction.When the parameter anti-predator behaviour is  = 0.473 to produce a stable system, then it means that the predator population has experienced demise.

Conclusion
2 for the capture rate , and continue their growth with conversion rate .Parameter  represents environmental carrying capacity,  is the maximum per capita prey growth rate,  is a mortality rate of a predator in the absence of prey,  namely the level of predator saturation, and  anti-predator behavior.

Figure 5 .Figure 7 .Figure 6
Figure 5. Stable Time Series Graphs at  2 with Initial Values[1,17] Construction of a prey-predator model with a Holling type III response function and anti-predator behavior is. 2 −  − .

Table 1 .
System Parameter Value.