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We give the first proof of the existence of a renormalisation fixed-point for period-doubling in pairs of maps of two variables lying in the so-called Feigenbaum-Summation (FS) universality class. The first map represents a subsystem that is unimodal with an extremum of degree two. The dynamics of the second map accumulates an integral characteristic of the dynamics of the first, via a particular form of unidirectional coupling. We prove the existence of the corresponding renormalisation fixed point by rigorous computer-assisted means and gain tight rigorous bounds on the associated universal constants. Our work provides the first step in establishing rigorously the picture conjectured by Kuznetsov et al of the birth, from the FS-type fixed point, of the so-called C-type two-cycle via a period doubling in the dynamics of the renormalisation group transformation itself.

Prostate cancer is the fifth most common cause of death from cancer, and the second most common diagnosed cancer in men. In the last few years many mathematical models have been proposed to describe the dynamics of prostate cancer under treatment. So far one of the major challenges has been the development of mathematical models that would represent in vivo conditions and therefore be suitable for clinical applications, while being mathematically treatable. In this paper, we take a step in this direction, by proposing a nonlinear distributed-delay dynamical system that explores neuroendocrine transdifferentiation in human prostate cancer in vivo. Sufficient conditions for the existence and the stability of a tumour-present equilibrium are given, and the occurrence of a Hopf bifurcation is proven for a uniform delay distribution. Numerical simulations are provided to explore differences in behaviour for uniform and exponential delay distributions. The results suggest that the choice of the delay distribution is key in defining the dynamics of the system and in determining the conditions for the onset of oscillations following a switch in the stability of the tumour-present equilibrium.

Using a hybrid cellular automaton with stochastic elements, we investigate the effectiveness of multiple drug therapies on prostate cancer (PCa) growth. The ability of Androgen Deprivation Therapy to reduce PCa growth represents a milestone in prostate cancer treatment, nonetheless most patients eventually become refractory and develop castration-resistant prostate cancer. In recent years, a "second generation" drug called enzalutamide has been used to treat advanced PCa, or patients already exposed to chemotherapy that stopped responding to it. However, tumour resistance to enzalutamide is not well understood, and in this context, preclinical models and in silico experiments (numerical simulations) are key to understanding the mechanisms of resistance and to assessing therapeutic settings that may delay or prevent the onset of resistance. In our mathematical system, we incorporate cell phenotype switching to model the development of increased drug resistance, and consider the effect of the micro-environment dynamics on necrosis and apoptosis of the tumour cells. The therapeutic strategies that we explore include using a single drug (enzalutamide), and drug combinations (enzalutamide and everolimus or cabazitaxel) with different treatment schedules. Our results highlight the effectiveness of alternating therapies, especially alternating enzalutamide and cabazitaxel over a year, and a comparison is made with data taken from TRAMP mice to verify our findings.

Prostate cancer represents the second most common cancer diagnosed in men and the fifth most common cause of death from cancer worldwide. In this paper, we consider a nonlinear mathematical model exploring the role of neuroendocrine transdifferentiation in human prostate cancer cell dynamics. Sufficient conditions are given for both the biological relevance of the model’s solutions and for the existence of its equilibria. By means of a suitable Liapunov functional the global asymptotic stability of the tumour-free equilibrium is proven, and through the use of sensitivity and bifurcation analyses we identify the parameters responsible for the occurrence of Hopf and saddle-node bifurcations. Numerical simulations are provided highlighting the behaviour discovered, and the results are discussed together with possible improvements to the model.

We gain tight rigorous bounds on the renormalization fixed point for period doubling in families of unimodal maps with degree 4 critical point. We use a contraction mapping argument to bound essential eigenfunctions and eigenvalues for the linearization of the operator and for the operator controlling the scaling of added noise. Multi-precision arithmetic with rigorous directed rounding is used to bound operations in a space of analytic functions yielding tight bounds on power series coefficients and universal constants to over 320 significant figures.