5D model of pancreatic cancer: Key features of ultimate dynamics

Abstract

In this paper, we study various types of ultimate dynamics for the 5-dimensional pancreatic cancer model without treatment. This model has been created by Hu et al. and describes interactions between pancreatic cancer cells (PCCs); pancreatic stellate cells (PSCs); immune cells and two types of cytokines, tumour–promoting and tumour–suppressing. Using iteratively the localization theorem of compact invariant sets we derive ultimate upper bounds for all cell populations and establish the property of the existence of the attracting set. Next, we find several conditions under which our system demonstrates various types of convergence dynamics: to equilibrium points located in the intersection of PCC-free plane with the PSC-free plane; equilibrium points with positive values of PSC density located in PCC-free plane; ω-limit sets with positive values of PCC density. Further, we explore the dynamics within the PSC-free plane and find attraction conditions to ω-limit sets with positive values of PCC density. Our assertions are expressed by means of iterative sequence of algebraic inequalities respecting parameters of the model. Our theoretical studies are supplied by results of numerical simulation. Locations of attracting equilibrium points have been found with help of the iteration procedure containing from 2 steps up to 23 steps in the dependence of parameter conditions and the location with respect to the PSC-free plane.