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We investigate the competition between two groups of similar agents in the restricted, but classical context of unstructured populations varying in continuous time in an isolated, homogeneous, and constant abiotic environment. Individual behavioral and phenotypic traits are quantified by one-dimensional strategies and intra- as well as inter-specific interactions are described in the vicinity of a stationary regime. Some known results are revisited: invasion by a new strategy generically implies the substitution of the former resident; and resident-invader coexistence is possible close to singular strategies-the stationary points of the invasion fitness-and is generically protected-each of the two competing groups can invade the other. An (almost known) old conjecture is shown true: competition close to a singular strategy is "essentially Lotka-Volterra"-dominance of one strategy, protected coexistence at an intermediate equilibrium, and mutual exclusion are the generic outcomes. And the unfolding of the competition scenarios is completed with the analysis of three degenerate singular strategies-characterized by vanishing second-order fitness derivatives-near which resident-invader coexistence can be unprotected. Our approach is based on the series expansion of a generic demographic model, w.r.t. the small strategy difference between the two competing groups, and on known results on time-scale separation and bifurcation theories. The analysis is carried out up to third order and is extendable to any order. For each order, explicit genericity conditions under which higher orders can be neglected are derived and, interestingly, they are known prior to invasion. An important result is that degeneracies up to third-order are required to have more than one stable way of coexistence. Such degeneracies can be due to particular symmetries in the model formulation, and breaking the genericity conditions provides a direct way to draw biological interpretations. The developed body of theory is exemplified on a model for the evolution of cannibalism and on Lotka-Volterra competition models., (Copyright © 2015 Elsevier Ltd. All rights reserved.)

Simple Summary: Neuroblastoma is a rare type of cancer that usually affects children. The high-risk patients' expected survival rate is less than 50%. One reason is the lack of precision in the standard treatment protocol: a one-size-fits-all multi-modal therapy. The study presented in this paper was designed to address this deficit by optimising the use of two chemotherapeutic agents—vincristine and cyclophosphamide—during induction chemotherapy—the part of the protocol that shrinks the primary tumour before surgical removal. We combined a mathematical model and an optimisation algorithm to identify the best chemotherapy schedules for a cohort of virtual patients with different initial tumour compositions. Our results reveal novel strategies to exploit a pair of drugs with different levels of efficacy, provide a platform on which to individualise induction chemotherapy, and lay the foundation for a personalised therapy that leverages targeted therapies, multi-region sequencing, liquid biopsies, and modern computational methods to improve today's multi-modal therapy. Neuroblastoma is the most common extra-cranial solid tumour in children. Despite multi-modal therapy, over half of the high-risk patients will succumb. One contributing factor is the one-size-fits-all nature of multi-modal therapy. For example, during the first step (induction chemotherapy), the standard regimen (rapid COJEC) administers fixed doses of chemotherapeutic agents in eight two-week cycles. Perhaps because of differences in resistance, this standard regimen results in highly heterogeneous outcomes in different tumours. In this study, we formulated a mathematical model comprising ordinary differential equations. The equations describe the clonal evolution within a neuroblastoma tumour being treated with vincristine and cyclophosphamide, which are used in the rapid COJEC regimen, including genetically conferred and phenotypic drug resistance. The equations also describe the agents' pharmacokinetics. We devised an optimisation algorithm to find the best chemotherapy schedules for tumours with different pre-treatment clonal compositions. The optimised chemotherapy schedules exploit the cytotoxic difference between the two drugs and intra-tumoural clonal competition to shrink the tumours as much as possible during induction chemotherapy and before surgical removal. They indicate that induction chemotherapy can be improved by finding and using personalised schedules. More broadly, we propose that the overall multi-modal therapy can be enhanced by employing targeted therapies against the mutations and oncogenic pathways enriched and activated by the chemotherapeutic agents. To translate the proposed personalised multi-modal therapy into clinical use, patient-specific model calibration and treatment optimisation are necessary. This entails a decision support system informed by emerging medical technologies such as multi-region sequencing and liquid biopsies. The results and tools presented in this paper could be the foundation of this decision support system. [ABSTRACT FROM AUTHOR]

We present in this paper the first example of chaotic evolutionary dynamics in biology. We consider a Lotka-Volterra tritrophic food chain composed of a resource, its consumer, and a predator species, each characterized by a single adaptive phenotypic trait, and we show that for suitable modeling and parameter choices the evolutionary trajectories approach a strange attractor in the three-dimensional trait space. The study is performed through the bifurcation analysis of the so-called canonical equation of Adaptive Dynamics, the most appropriate modeling approach to long-term evolutionary dynamics. [ABSTRACT FROM AUTHOR]

The paper describes the slow evolution of two adaptive traits that regulate the interactions between two mutualistic populations (e.g. a flowering plant and its insect pollinator). For frozen values of the traits, the two populations can either coexist or go extinct. The values of the traits for which populations extinction is guaranteed are therefore of no interest from an evolutionary point of view. In other words, the evolutionary dynamics must be studied only in a viable subset of trait space, which is bounded due to the physiological cost of extreme trait values. Thus, evolutionary dynamics experience so-called border collision bifurcations, when a system invariant in trait space hits the border of the viable subset. The unfolding of standard and border collision bifurcations with respect to two parameters of biological interest is presented. The algebraic and boundary-value problems characterizing the border collision bifurcations are described together with some details concerning their computation. [ABSTRACT FROM AUTHOR]