Topological Complexity and Predictability in the Dynamics of a Tumor Growth Model with Shilnikov's Chaos

Dynamical systems modeling tumor growth have been investigated to determine the dynamics between tumor and healthy cells. Recent theoretical investigations indicate that these interactions may lead to different dynamical outcomes, in particular to homoclinic chaos. In the present study, we analyze both topological and dynamical properties of a recently characterized chaotic attractor governing the dynamics of tumor cells interacting with healthy tissue cells and effector cells of the immune system. By using the theory of symbolic dynamics, we first characterize the topological entropy and the parameter space ordering of kneading sequences from one-dimensional iterated maps identified in the dynamics, focusing on the effects of inactivation interactions between both effector and tumor cells. The previous analyses are complemented with the computation of the spectrum of Lyapunov exponents, the fractal dimension and the predictability of the chaotic attractors. Our results show that the inactivation rate of effector cells by the tumor cells has an important effect on the dynamics of the system. The increase of effector cells inactivation involves an inverse Feigenbaum (i.e. period-halving bifurcation) scenario, which results in the stabilization of the dynamics and in an increase of dynamics predictability. Our analyses also reveal that, at low inactivation rates of effector cells, tumor cells undergo strong, chaotic fluctuations, with the dynamics being highly unpredictable. Our findings are discussed in the context of tumor cells potential viability

A dinâmica do modelo populacional de Ricker

O modelo de Ricker é um dos vários modelos utilizados em ecologia para descrever a dinâmica populacional que depende da densidade ao longo do tempo, observada em intervalos de tempo discreto. Este modelo é adequado para estudar espécies onde as estimativas anuais (ou geracionais) de abundância são caracterizadas adequadamente como dinâmica populacional e para transições entre estágios da história de vida, tais como produção de descendentes ou sobrevivência de um estágio para o próximo. A última aplicação é comum na ciência da pesca, onde o modelo de Ricker é frequentemente usado para relacionar a produção de recrutas (peixes jovens que sobrevivem para se juntar à população) a fatores na densidade observada, tais como abundância, biomassa total ou potencial total de desova de peixes adultos, como parte de um modelo populacional abrangente. O objetivo deste trabalho é aplicar alguns resultados da teoria de estabilidade de sistemas dinâmicos discretos ao Modelo de Ricker. Em particular, estudamos a dinâmica populacional em ambos os casos, no modelo autónomo e no modelo não autónomo.The Ricker model is one of several models used in ecology to describe population dynamics that depend on density over time, observed at discrete intervals. This model is suitable for studying species where annual (or generational) estimates of abundance are adequately characterized as population dynamics, and for transitions between life history stages, such as offspring production or survival from one stage to the next. The latter application is common in fisheries science, where the Ricker model is frequently used to relate the production of recruits (young fish that survive to join the population) to factors in observed density, such as abundance, total biomass, or total potential spawning of adult fish, as part of a comprehensive population model. The objective of this work is to apply some results from the theory of stability of discrete dynamic systems in the Ricker model. In particular, we study population dynamics in both autonomous and non-autonomous cases