Stability Analysis of Prey-Predator Model With Holling Type IV Functional Response and Infectious Predator

Stability of equilibrium points of the prey-predator model with diseases that spreads in predators where the predation function follows the simplified Holling type IV functional response are investigated. To find out the local stability of the equilibrium point of the model, the system is then linearized around the equilibrium point using the Jacobian matrix method, and stability of the equilibrium point is determined via the eigenvalues method. There exists three non-negative equilibrium points, except , that may exist and stable. Simulation results show that with the variation of several parameter values infection rate of disease , the diseases in the system may become endemic, or may become free from endemic.


INTRODUCTION
Many researchers have studied the dynamics of populations using mathematical models. The constructed model based on data and real phenomena is believed can be used to predict and give more information qualitatively. Mathematical models related to natural resources and management can be seen for examples in [10], [11], and [12]. Besides that, the mathematical models have also been widely used in investigating the behavior of diseases spread, for example see [3], [8] and [13]. In ecological problems, species populations are very important to study their existence. The existence of the populations is influenced by many factors, two of which are ecological and epidemiological factors. The ecological factors include Intra and interspecies interaction in the form of competition and predation, and epidemiological factors include the spread of infectious disease [7].
Several previous researchers were interested to study the effects of the disease in the predatorprey system. Foryś and Radziński [6] analyzed the diffusive predator-prey system with disease in predator species. They showed that the dynamics of the system mainly depend on the infection rate parameter . When the value of is small, there is no positive equilibrium and they expect that the 156 A. Muh. Amil Siddik, Syamsuddin Toaha, Andi Muhammad Anwar Jurnal Matematika, Statistika & Komputasi population of infected predators will become extinct. The predator-prey model with parasitic infections that spread only to predator populations in the form of SIS (Susceptible Infected Susceptible) has been reviewed by Haque [5]. The analysis showed that infection with predators can save prey populations from extinction. From the sequence of studies, it can be illustrated that the infection rate affects the prey can still exist in the system.
Besides the spread of disease, interactions between species in an ecosystem cause the state of a species populations can change. These interactions can have a positive, negative or even no effect in the species that interact. To express these phenomena, several researchers tried to understand using an appropriate mathematical method. A suitable mathematical model in ecology cannot be achieved easily because this model could never predict nature's behavior accurately. In facing this major problem, we can search for analyzable model that can be close enough to reality [1].
Andrews proposed a response called the Monod-Haldane functional response, see [2].
Collings used this function and called it Holling type IV response, see [4]. This response function describes a situation in which the predator's per capita rate of predation decreases at sufficiently high prey densities. Sokol and Howell [9] suggested a simplified Holling type IV function of the form and found that it is simpler and better than the original function of Holling type IV.
The study aims to determine the effect of the disease in predators to populations in the system, where the predator functional response is based on the simplified Holling type IV. This study using simplified Holling type IV because some investigations about the prey-predators system with the disease, using Holling type II functional response, while few investigations using Holling type IV functional response in population ecology, see [5], [6], and [7]. Furthermore, the conditions for the existence of positive equilibrium points and their stability are analyzed.

Assumptions
The following assumptions are considered in formulating the model: (1) In the presence of disease, the predator population is divided into two classes, namely susceptible predator and infected predator at time t denoted by and respectively. Therefore, at time t the total predator population is denoted as .
(2) The disease spreads among the predator population only through in-group and out-group interactions and is not genetically inherited. The infected population do not recover or become immune. The incidence function is assumed to be nonlinear function , which is a standard incidence.

Model Formulation
The model consisting of prey population density at time which is denoted by . The susceptible predator population density at time is denoted by and the infected predator population density at time is denoted by . The total population of the predator at time is denoted by . The susceptible and infected predators prey on the food according to the simplified Holling type IV functional response.
The dynamics of the population model can be represented in the form of the differential equations system as the following: The constants in the model (1) are assumed to be positive. The constant is the growth rate of prey . Constant represents the intraspecific competition among individuals of prey . Constant is the half-saturation constant, and is the predator's natural mortality rate. Constants is the maximum value which per capita reduction rate can attain. Constants has a similar meaning to . Constant is a disease standard incidence disease-induced mortality rate of infected predators. Let denote the fractions of susceptible predators and denote the fractions of infected predators. If , , then The system (1) is then reduced in the form

Boundedness
All of the parameters of the system (2) are non-negative, and so the corresponding right-hand side of the system is a smooth function of the variables in the region . It follows that local existence and uniqueness properties hold for the solution of the system. Any solution with initial values eventually approaches this region.

Equilibrium points
The equilibrium points in which prey is present are studied only. There are several nonnegative equilibrium points that can exist. a

Stability Analysis of Equilibrium Points
The Jacobian matrix for the system (2)

Numerical Simulations
In this section, the dynamical of system (2) is studied and visualized numerically. From the suitable condition of existence equilibrium points, we set following fixed parameters values with appropriate units (5) Using initial value and The dynamical of population for some values of infection rate are done using numerical method rkf45, and system (2) is running for 500 time steps.
The first case, we assume that half of the populations become infected. So . (fraction of infected predator = 1), so it means that the system (2) becomes endemic.

When
, and can be exist. The equilibrium point is not stable, is not stable, and is stable. It means that the system (2) becomes endemic.
The second case, we use and , but not too small. A. Muh. Amil Siddik, Syamsuddin Toaha, Andi Muhammad Anwar

Jurnal Matematika, Statistika & Komputasi
From the second case, we get not stable, not stable, and stable. It means that the system (2) becomes endemic. This case same as first case. The difference is only seen in the oscillation of the system. See  There are some interesting things that may happen in the third case, when we use small value of , say . We get is not stable, is stable, and is not stable. It means that the system (2) becomes free from endemic with the predators species can still exist, see Figure 3.3. This result is different from Foryś and Radziński  (5) and . (B) 3D of the system (2) stable points with data set (5) and