The diffusive Beddington-DeAngelis predator-prey model with nonlinear prey-taxis and free boundary Norov Abdurahmon Qiyomiddin o’g’li, junior researcher

The diffusive Beddington-DeAngelis predator-prey model with nonlinear prey-taxis and free boundary
Norov Abdurahmon Qiyomiddin o’g’li, junior researcher,
[e-mail],
Institute of Mathematics named after V.I.Romanovskiy, Tashkent, Uzbekistan
Abstract-The diffusive Beddington-DeAngelis predator-prey model with nonlinear prey-taxis and free boundary is considered. For the solutions of the problem apriori estimates are established. On the base of apriori estimations the existence and uniqueness of theorems are proved.
In this work, we consider the free boundary problem for a system of parabolic reaction-diffusion equations. It is required to find functions , , in the domain satisfying the conditions
(1)
where –free boundary, and represent predator and prey densities, respectively. Constants are positive; is the mortality rate of the predator, which does not depend on the prey density; the function is the Beddington-DeAngelis functional response; and is the conversion rate from prey to predator.
In this model, the predator is attracted by the prey and denotes the prey-tactic sensitivity and satisfies, for some positive constants and , for , and is Lipschitz continuous, ie, for
Throughout this paper, we assume that and initial functions satisfy

The problem (1) has been studied in [1] for . We extend [1] and some results of the predator and prey model.
Theorem. Let , , be a solution of (1). Then

.
,
References

1. 
   Wang J, Wang M. The diffusive Beddington-DeAngelis predator-prey model with nonlinear prey-taxis and free boundary. Math Meth Appl Sci. 2018; 1–22.
   
2. 
   Wang M, Zhang Y. Two kinds of free boundary problems for the diffusive prey-predator model. Nonlinear Anal Real World Appl. 2015; 24:73-82.