Abstract
In this paper, a mathematical model consisting of the prey-predator model, prey is at risk of disease then become as susceptible and infected, while predator with different stage structure: immature and mature predator, the infected prey is at risk recover and harvest. The function of disease is proportionality function. At the beginning, the reasons of studying stage structure and the dynamic of nontrivial subsystems that may be lead to coexistence of these types of spices explain and by using Maple software, Jacobean matrix, Routh-Hurwitz criterion, Bendixson-Dulac criterion and Lyapunov function to prove the existence, periodic solution, local and global stability. We concluded that the survival for two preys are possible through the non-periodic solution due to the Bendixson-Dulac criterion, also the immature predator neither attack preys nor yield offspring's and die when the mature predator extinction, the global stability conditions for the original system be stretch of global stability conditions for subsystems. Finally, Mathematica software employs to describe some results in numerical simulation
 
 http://dx.doi.org/10.25130/tjps.25.2020.040
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