Predator-prey model with a free boundary

Predator-prey model with a free boundary
Elmurodov A.N.
Institute of Mathematics, Tashkent, Uzbekistan;
e-mail: [e-mail]
In this article, we consider the following Beddington-DeAngelis and Tanner reaction-diffusion system of predator-prey model:
(1)
where is called a free boundary, which is to be determined together with all the parameters , and are given positive constants.
The problem of describing the dynamic process of the invasion of a new competitor into the habitat of a local species belongs to [1, 2], who investigated the nonlinear evolution of two species in an unbounded spatial domain.
There are the Beddington-DeAngelis and Tanner type functional response contained in the first and second equation of model (1), respectively, where and represent the population density of the predator and the prey at time with diffusion rates and , respectively. We suppose that the two diffusion rate are positive and equal, but not necessary constants.
This model describes how a new species with population density invades into the habitat of a native competitor .
Since the natural biological process is associated with the invasion, we consider it necessary to add terms reflecting the transfer and change in concentration to the equations using the transfer term and the new “bio capacity” coefficient.
For species that inhabit a terminal region with an outer boundary death point . However, the evolution equation satisfied by is valid for the developing domain . The free boundary condition, which is well known as the Stefan condition, states that the rate at which the free boundary expands is proportional to the population gradient at that location.
Problem (1) with , , was studied in [1].
The coefficients , and functions , satisfy:

The main contribution of this article is the establishment of the global existence of the classical solution to problem (1) and the study of the behavior of the solution. A method is proposed for establishing a priori Schauder-type estimates for a new class of free boundary problems for mixed-two-phase equations. The principle of comparison is proved.
R E F E R E N C E S
1. Liu Yu., Guo Z., Smaily El M. and Wang L. Biological invasion in a predator–prey model with a free boundary. Boundary Value Problems. (2019) 2019:33. https://doi.org/10.1186/s13661-019-1147-7
2. Shi H. B., Li Y.. Global asymptotic stability of a diffusive predator-prey model with ratio-dependent functional response, Appl. Math. Comput. 25(2015) 71–77.