In spatial ecology, the concept of resource selection expresses the idea that for many animals, the distribution of an individual's location is not uniform over the region available to them; instead, they spend time preferentially in some locations compared to others, in a way that can often be related to spatial covariates. The related concept of step selection describes the variation in an individual's tendency to move to particular locations in the short term, taking into account both spatial covariates and the constraints of their process of movement. Consistent modelling of resource selection and step selection is necessary to understand animals' distribution in space and to interpret movement, telemetry, and spatial survey data in a meaningful way. In this paper, I take advantage of recent developments in stochastic processes and statistical algorithms to develop a range of new stochastic models in which both the dynamics and the long-term behaviour are tractable and described parametrically, and which are flexible enough to represent a wide range of patterns of movement and space use encountered in reality. I extend the mathematical analogy between movement modelling and Markov chain Monte Carlo algorithms, first proposed by Michelot, Blackwell & Matthiopoulos (2019; Ecology 100, e02452), to a wide range of continuous-time stochastic processes, including both diffusion processes and velocity-jump models, that in different ways are motivated by the simple discrete-time step-and-turn models widely used in practice. Particular cases include a diffusion process where the dynamics are defined in terms of speed and direction of movement, and a velocity-jump process in d dimensions, generalizing the 'bouncy particle sampler' used in Bayesian inference, in which the distribution of velocity after a so-called 'bounce' event has support over a region which itself has dimension d. I also show how this mathematical approach can be extended to models incorporating distinct behavioural states and to higher dimensional models representing the joint movement of interacting individuals.

This topical collection of the Journal of Mathematical Biology is dedicated to Professor Ezio Venturino on the occasion of his 70th birthday. It brings together recent advances in the mathematical modelling and analysis of biological systems, spanning topics from dynamical systems and stochastic processes to applications in ecology, epidemiology, evolutionary dynamics, and biomedicine. The contributions highlight both new theoretical developments and innovative computational techniques, reflecting the vitality of current research in mathematical biology and offering a fitting tribute to Professor Venturino's influential scientific career.

This paper investigates the conditions for the stability and emergence of patterns in a new three-component reaction-diffusion system. The system describes the coexistence and interaction of water reservoirs, vegetation, and bushfire activity in a given ecosystem. We perform a detailed stability analysis to determine the parameter space where an unstable homogeneous equilibrium becomes stable with respect to spatially nonuniform perturbations. We also use diffusion to generate traveling trains in the form of periodic orbits of the linearized system. These orbits are remnants of an unstable equilibrium in the absence of diffusion and arise from a nonsingular eigenvalue crossing of the imaginary axis, while a third eigenvalue remains real and negative, thereby ensuring linear stability for monocromatic waves. These phenomena differ from "classical" Turing and Hopf bifurcations, as the model does not involve distinct "activators" and "inhibitors", and the effects observed are not the byproduct of diffusion with necessarily differing speeds. Also, differently from the classical Turing pattern, the role of diffusion in this context is to stabilize, rather than destabilize, homogeneous equilibria. We also consider the case of plant competition, showing a suitable form of Turing instability for slow-frequency oscillations in a small rainfall regime.

In river ecosystems, unidirectional flow and habitat fragmentation threaten prey persistence, necessitating protection zones. This study explores the dynamics of a generalist predator-prey model with a boundary-upstream protection zone for prey and Holling-II type functional response in open advective environments. Firstly, we establish complete system dynamics driven by the interplay of variable advection rates, protection zone lengths and predator carrying capacity, revealing their coupled effects on population persistence and extinction. We identify a critical protection zone length, beyond which predator take-over is prevented within a prey advection range, regardless of predator strength, thereby promoting species coexistence. Furthermore, through bifurcation analysis, the existence and multiplicity for positive steady states are studied, linking prey attack-evasion ability and predator advection rate to bistable outcomes. The asymptotic behaviors of positive steady states induced by large diffusion rates are characterized, showing that enhanced dispersal counteracts flow-induced washout. This study uniquely integrates Holling-II response with upstream refuge effects, revealing critical thresholds for predator control. The findings guide designing flow-adaptive protection zones to mitigate generalist predator dominance in aquatic ecosystems, offering actionable management strategies.

Ordered leaf attachment, Phylo2Vec, and HOP are three recently introduced vector representations for rooted phylogenetic trees where the representation is determined by an ordering of the underlying leaf set X. Comparing the vectors of two rooted phylogenetic X-trees T and for a fixed ordering on X leads to polynomial-time computable measure for the dissimilarity of T and , albeit dependent on the choice of the leaf ordering. For each of ordered leaf attachment, Phylo2Vec, and HOP, we compare this measure with the rooted subtree prune and regraft distance (rSPR), the hybrid number, and the temporal tree-child hybrid number of T and . Although there is no direct relationship between rSPR and any of the three vector-based measures, we show that, when minimized over all orderings, the hybrid number is equal to HOP and is an upper bound on the other two. Moreover, when minimized over all orderings induced by common cherry-picking sequences of T and , the temporal tree-child hybrid number of T and is equal to each of the three vector-based measures.

Animals continuously update their movement decisions using both real-time observations and historical information from experience, social interactions, or environmental cues, which we call accumulated memory (also called distributed delay memory). While memory is important for animals, how it influences movement strategies has received limited attention. We address this gap by integrating accumulated memory into three widely used models: advection-diffusion, Fickian-type diffusion, and Fokker-Planck type diffusion. These represent distinct strategies: (i) gradient-based movement, responding to environmental gradients; (ii) environment matching, symmetrically adjusting movement rates; and (iii) location-based movement, relying solely on local suitability. We derive each model from random walk models to compare how different memory-based movement strategies at the individual level give rise to distinct macroscopic population behaviors. Furthermore, we establish the local existence of solutions for a general model encompassing all three cases using fixed-point theory and provide a linear stability analysis. Numerical simulations show that the Fickian model always converges rapidly to a uniform state. Under memory-suppressed conditions, the advection-diffusion and Fokker-Planck models may exhibit aggregation, whereas under memory-enhanced conditions all models eventually reach uniformity, with the advection-diffusion and Fokker-Planck models sometimes displaying oscillatory wiggling pattern or periodic movement before stabilization.

We consider the impact of network connectivity on the dynamics of a population in a stream environment. The population is modeled using a graph theoretical framework, with habitats represented by isolated patches. We introduce a change in connectivity into the model through the addition of a bi-directional or one-directional edge between two patches and examine the impact of this edge modification on the metapopulation growth rate and the network biomass. Our main results indicate that adding a bi-directional edge often decreases both measures, while the effect of adding one-directional edge is more intricate and dependent on the model parameters. We establish complete analytical results for stream networks of three patches, and provide some generalizations and conjectures for more general stream networks of n patches. These conjectures are supported with numerical simulations.

This paper presents a novel West Nile virus model that has more extensive free boundary conditions and also takes into account the impact of infected mosquitoes on the free boundary, both of which are firsts in West Nile virus modeling. Specifically, the free boundary conditions independent of the dispersal kernel functions in the equations, bring new challenges to the dynamical analysis of spreading-vanishing, especially for the case where the basic reproduction number , which involves new ideas and techniques for dynamics analysis. Moreover, due to the consideration of the impact of infected mosquitoes in the free boundary conditions, new conclusions have been obtained. Numerical schemes have been developed, which not only verify qualitative theoretical results, but also provide novel quantitative insights into the effects of various factors on transmission dynamics. Overall, our results not only differ significantly from the local diffusion version presented in Lin and Zhu (2017) but also extend all the conclusions from the nonlocal diffusion version in Du and Ni (2020), with some conclusions obtained under more general conditions.

Qualitative properties of adequate numerical methods for stochastic differential equations to simulate the stochastic counterparts of some known models with variable contact diffusion rates in mathematical epidemiology are investigated, especially SIS and SIR models with non-constant total population sizes on bounded domains. We show the convergence of numerical approximations based on Balanced Implicit Methods (BIMs) by proving several properties; including positivity, invariance, stability, mean and mean square consistency, local uniform boundedness, mean square Hölder continuity and mean square contractivity. We also present some simulation results for those models with realistic parameters.