Anaysis of Predator-Prey Dynamics Using Holling Type I & II Response Functions with Kleptoparasitism and Anti-Predator Behavior

Abstract

Predator-prey interactions involving 3 species in an African forest ecosystem between Deer, Hyena and Lion considering the influence of kleptoparasitism and anti-predator behaviour using Holling type I $\&$ II functional responses. This predator prey model is constructed based on the assumption that the behaviour of the second predator Hyena often has the ability to defend itself against other predators such as fleeing, fighting, and intimidating which is called anti-predator behaviour. Based on the existing phenonema, the objectives of this study are to determine the model construction, equilibrium point analysis and stability, as well as numerical simulation and interpretation of the prey-prey model using Holling type I $\&$ II functional responses in the presence of kleptoparasitism and anti-predator behaviour. The calculation analysis in this study was carried out by finding the equilibrium point and stability analysis. The results of the dynamic analysis show that there are five equilibrium points with the type of stability, namely $E_1(x, y, z) = (0, 0, 0)$ which states the extinction of the three populations, point equilibrium $E_2(x, y, z) = (K, 0, 0)$ which represents the extinction of the first predator and second predator populations, point equilibrium $E_3(x, y, z)$ $=$ $\left(-\frac{\theta_2}{\theta_2b-\mu_2},0,-\frac{r\mu_2(K(\beta\theta_2-\mu_2)+\theta_2)}{K\beta_2(b\theta_2-\mu)^2}\right)$ which expresses extinction in the first predator population, point equilibrium $E_4(x,y,z) = \left(\frac{\theta_1}{\mu_1},\frac{r(\mu_1K-\theta_1)}{\mu_1K\beta_1},0\right)$ which expresses extinction at the second predator, and equilibrium point $E_5(x^*,y^*,z^*)$ which states that all three populations can coexist. Numerical simulation results show the existence of double stability at points $E_4$ and $E_6$ when the parameter values $\mu_1 = 0.3, \mu_2 = 0.12$ and double stability occurs again at points $E_4$ and $E_3$ when the parameter variation values $\mu_1 = 0.3, \mu_2 = 0.158$.