A population model of two-strains tumors with piecewise constant arguments

In this study, the population growth of the brain tumor GBM, is constructed such as( )( ( ) ( [ ] )) ( ) ( [ ] ) ( ) ( [ ] )( )( ( ) ( [ ] )) ( [ ] ) ( ) [ ]1 1 1 2 1 12 2 1 2 1 2d = ( ) rx t R x t x t x t y t d x t x tdd =r y t R y t y t x t y t d y( ) ( t )dx px tty t ytα α γβ β γ⎧ + − − ⎡ ⎤ − ⎡ ⎤ − ⎡ ⎤ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎪⎪⎨⎪− − ⎡⎣ ⎤⎦ + ⎡⎣ ⎤⎦ − ⎡⎣ ⎤⎦ ⎪⎩(A)where t = 0 , the parameters 1 2 1 2 1 1 2 1 2 1 α ,α , β , β ,γ , p, d , d ,R , R , r and 2 r are positivereal numbers and [[t]] denotes the integer part of t∈[0,8) . System (A) explains a tumorgrowth, that produces after a specific time another tumor population with different growth rateand different treatment susceptibilities. The local and global stability of this model is analyzedby using the theory of differential and difference equations. Simulations and data of GBM givea detailed description of system (A) at the end of the paper.