Bulletin of Mathematical Biology最新文献

We study pattern formation in a chemotaxis model of bacteria and soil carbon dynamics as an example system where transient dynamics can give rise to pattern formation outside of Turing unstable regimes. We use a detailed analysis of the reactivity of the non-spatial and spatial dynamics, stability analyses, and numerical continuation to uncover detailed aspects of this system's pattern-forming potential. In addition to patterning in Turing unstable parameter regimes, reactivity of the spatial system can itself lead to a range of parameters where a spatially uniform state is asymptotically stable, but exhibits transient growth that can induce pattern formation. We show that this occurs in the bistable region of a subcritical Turing bifurcation. Intriguingly, such bistable regions appear in two spatial dimensions, but not in a one-dimensional domain, suggesting important interplays between geometry, transient growth, and the emergence of multistable patterns. We discuss the implications of our analysis for the bacterial soil organic carbon system, as well as for reaction-transport modeling more generally.

Cell heterogeneity plays an important role in patient responses to drug treatments. In many cancers, it is associated with poor treatment outcomes. Many modern drug combination therapies aim to exploit cell heterogeneity, but determining how to optimise responses from heterogeneous cell populations while accounting for multi-drug synergies remains a challenge. In this work, we introduce and analyse a general optimal control framework that can be used to model the treatment response of multiple cell populations that are treated with multiple drugs that mutually interact. In this framework, we model the effect of multiple drugs on the cell populations using a system of coupled semi-linear ordinary differential equations and derive general results for the optimal solutions. We then apply this framework to three canonical examples and discuss the wider question of how to relate mathematical optimality to clinically observable outcomes, introducing a systematic approach to propose qualitatively different classes of drug dosing inspired by optimal control.

The linear framework is an approach to analysing biochemical systems based on directed graphs with labelled edges. When applied to individual molecular systems, graph vertices correspond to system states, directed edges to transitions, and edge labels to transition rates. Such a graph specifies the infinitesimal generator of a continuous-time Markov process. The master equation of this Markov process, which describes the forward evolution of vertex probabilities, is a linear differential equation, after which the framework is named, whose operator is the Laplacian matrix of the graph. The Matrix-Tree theorem, when applied to this Laplacian matrix, allows the steady-state probabilities of the Markov process to be expressed as rational algebraic functions of the transition rates. This capability gives algebraic access to problems that have otherwise been treated by approximations or numerical simulations, and enables theorems to be proved about biochemical systems that rise above their underlying molecular complexity. Here, we extend this capability from the steady state to the transient regime. We use the All-Minors Matrix-Tree theorem to express the moments of the conditional first-passage time distribution, and the corresponding splitting probabilities, as rational algebraic functions of the transition rates. This extended capability brings many new biological problems within the scope of the linear framework.

Chase-and-run dynamics, in which one population pursues another that flees from it, are found throughout nature, from predator-prey interactions in ecosystems to the collective motion of cells during development. Intriguingly, in many of these systems, the movement is not straight; instead, 'runners' veer off at an angle from their pursuers. This angled movement often exhibits a consistent left-right asymmetry, known as lateralisation or chirality. Inspired by such phenomena in zebrafish skin patterns and evasive animal motion, we explore how chirality shapes the emergence of patterns in nonlocal (integro-differential) advection-diffusion models. We extend such models to allow movement at arbitrary angles, uncovering a rich landscape of behaviours. We find that chirality can enhance pattern formation, suppress oscillations, and give rise to entirely new dynamical structures, such as rotating pulses of chasers and runners. We also uncover how chase-and-run dynamics can cause populations to mix or separate. Through linear stability analysis, we identify physical mechanisms that drive some of these effects, whilst also exposing striking limitations of this theory in capturing more complex dynamics. Our findings suggest that chirality could have roles in ecological and cellular patterning beyond simply breaking left-right symmetry.

The coexistence of species in predator-prey systems is a critical ecological issue due to the intricate interactions among multiple influencing factors. In this study, we develop a predator-prey model that incorporates prey herd behavior, cooperative hunting strategies among predators, and the establishment of a reserved area for prey protection. We establish conditions for the positivity and boundedness of the system to ensure long-term biological feasibility. The existence and stability of equilibrium points, along with the conditions for Hopf and saddle-node bifurcations, are rigorously analyzed. Numerical simulations are performed to validate the analytical findings. Global sensitivity analysis reveals that key parameters, including the size of the reserved area, predator cooperation, and migration rates, significantly affect system dynamics and species coexistence. Our numerical results suggest that expanding the reserved area promotes prey recovery, with predator populations initially growing but eventually declining towards extinction. Increased hunting cooperation among predators initially boosts predator populations but ultimately accelerates prey depletion, leading to predator collapse due to overhunting.

Social distancing is now a familiar strategy for managing disease outbreaks, but it is important to understand the interaction between disease dynamics and social behaviour. We distinguished the fully susceptibles from the social-distancing susceptibles and proposed a Filippov epidemic model to study the effect of social distancing on the spread and control of infectious diseases. The threshold policy is defined as follows: once the number of infected individuals exceeds the threshold value, social-distancing susceptibles take more stringent social-distancing practices, resulting in a decreasing infection rate. The target model exhibits novel dynamics: in addition to the coexistence of two attractors, it also demonstrates the coexistence of three attractors. In particular, bistability of the regular endemic equilibrium and the disease-free equilibrium occurs for the system; multistability of the regular endemic equilibrium, a pseudo-equilibrium and the disease-free equilibrium also occurs for the system. Discontinuity-induced bifurcations, including boundary-node, focus and saddle-node bifurcations, occur for the proposed model, which reveals that a small change in threshold values would significantly affect the outcome. Our findings indicate that for a proper threshold value, the infections can be ruled out or contained at the previously given level if the initial infection is relatively small.

It is generally recognized that oscillatory dynamics of mutualism systems arise from external factors such as environmental fluctuations and additional interspecific interactions. However, we here theoretically demonstrate that the saturating density dependence of mutualistic benefits and costs can lead to the periodic oscillations of obligate mutualism systems. This suggests that the dynamic complexity of mutualisms can also arise intrinsically. Our model differentiates benefits in mutualistic interactions from costs and assumes they respectively influence the reproduction rate and mortality of populations. In the symmetric case, where the model structure and parameters are the same for both species, this model shows multiple equilibria and oscillatory dynamics. The difference between benefit and cost may be the primary determinant of these phenomena. The system exhibits damped or periodic oscillations when this difference is intermediate. The two species can stably coexist when benefits significantly outweigh costs, whereas the system faces extinction when costs become relatively high. Asymmetry in benefit and cost between mutualists dramatically changes the system's dynamical regimes. Essentially, these oscillations of mutualism are caused by the transitions of the system between mutualism and antagonism. In addition, our model reveals the transient dynamics of the mutualism system (a phenomenon of regime shift without parameter change), including saddle crawl-bys (moving slowly by saddles) and ghost attractors (slow change in system state near the attractors). Our findings highlight the crucial role of nonlinear benefits and costs in the dynamical complexity of mutualisms.

We formulate a cell-scale model for the degradation of the extra-cellular matrix by membrane-bound and soluble matrix degrading enzymes produced by cancer cells. Based on the microscopic model and using tools from the theory of homogenisation we propose a macroscopic model for cancer cell invasion into the extra-cellular matrix mediated by bound and soluble matrix degrading enzymes. For suitable and biologically relevant initial data we prove the macroscopic model is well-posed. We propose a finite element method for the numerical approximation of the macroscopic model and report on simulation results illustrating the role of the bound and soluble enzymes in cancer invasion processes.

Many classes of phylogenetic networks have been proposed in the literature. A feature of several of these classes is that if one restricts a network in the class to a subset of its leaves, then the resulting network may no longer lie within this class. This has implications for their biological applicability, since some species - which are the leaves of an underlying evolutionary network - may be missing (e.g., they may have become extinct, or there are no data available for them) or we may simply wish to focus attention on a subset of the species. On the other hand, certain classes of networks are 'closed' when we restrict to subsets of leaves, such as (i) the classes of all phylogenetic networks or all phylogenetic trees; (ii) the classes of galled networks, simplicial networks, galled trees; and (iii) the classes of networks that have some parameter that is monotone-under-leaf-subsampling (e.g., the number of reticulations, height, etc.) bounded by some fixed value. It is easily shown that a closed subclass of phylogenetic trees is either all trees or a vanishingly small proportion of them (as the number of leaves grows). In this short paper, we explore whether this dichotomy phenomenon holds for other classes of phylogenetic networks, and their subclasses.