Shigui Ruan

In this paper, we revisit a honeybee-parasite interaction model with constant and periodic queen egg-laying rates, that is, in constant and periodic environments. In the constant environment, we obtain complete high codimension bifurcations, including a codimension-3 nilpotent saddle bifurcation and a codimension-2 Hopf bifurcation, and find that the nilpotent saddle of codimension 3 acts as the organizing center of the bifurcation sets. Our results indicate that the interacting outcomes (coextinction, survival of only honeybees, and coexistence) can be classified by a threshold value for the semisaturation constant of the honeybee population and two threshold values for the parasitism rate. In the periodic environment, we explore the bifurcation of a stable hyperbolic equilibrium (or a stable limit cycle) into an asymptotically stable periodic solution (or an attracting invariant torus). Numerical simulations are given to illustrate the theoretical results, including multitype bistability, tristability, existence of limit cycles, and invariant tori.

This paper is concerned with propagation phenomena in a nonlocal delayed reaction-diffusion equation in time almost periodic media. More specifically, we investigate the existence and nonexistence of transition waves and the upper and lower bound estimates of spreading speeds. By studying properties of exponential decaying solutions of the corresponding linearized problem, constructing appropriate upper and lower solutions, and using comparison arguments, we first prove that no matter the birth rate function is monotone or not, there is a critical wave speed such that a transition wave exists as soon as the mean value of the wave speed is greater than this critical speed. Moreover, in the monotone case we confirm the asymptotic behavior of the transition wave. Under some further assumptions, we establish the upper and lower bound estimates of spreading speeds for solutions of the Cauchy problem with slow and fast exponential decay initial values and also obtain the nonexistence of transition waves. Finally, a brief discussion is given to demonstrate that the critical wave speed obtained in the present paper coincides with the minimum speed and spreading speeds observed by others for some special cases.

Chikungunya (CHIK) is a viral disease transmitted to humans through the bites of Aedes mosquitoes infected with the chikungunya virus (CHIKV). CHIKV has been imported annually to Florida in the last decade due to Miami’s crucial location as a hub for international travel, particularly from Central and South America, including Brazil, where CHIK is endemic. This work addresses the geographic spread of CHIK, incorporating factors such as human movement, temperature dependency, as well as vertical transmission, and incubation periods, for different patches. Central to the model is the integration of a multipatch framework and the inclusion of human movement between endemic Brazilian states and Florida in the numerical analysis. We establish crucial correlations between the mosquito reproduction number [Formula: see text] and the disease reproduction number [Formula: see text] with the disease dynamics in a multipatch environment, encompassing not only from a theoretical perspective but also from numerical simulations. More specifically, after establishing the well-posedness of the multipatch model problem, we present conditions to ensure the uniform persistence of the disease. Next, using real population and temperature data we are able to determine all parameters that are temperature-dependent for both Brazil and Miami-Dade County and apply a two-patch model to calibrate the CHIK data from Brazil and Miami-Dade County. Finally, via sensitivity analysis we discuss the disease dynamics under different scenarios and make future projections, offering insights for potential effective control strategies, as well as addressing the timing for these strategies to be adopted.

Motivated by the invasion of Aedes albopictus mosquitoes and the interspecific competition between Aedes aegypti and Aedes albopictus in Florida, we formulate a two-species competition model on a compact metric graph. This model accounts for the species’ ability to inhabit and traverse along the graph’s edges. In the scenario of weak-strong competition, we prove that solutions of the competition model converge uniformly to a semi-positive equilibrium, where either Aedes albopictus survives while Aedes aegypti goes extinct, or vice versa. Whereas for weak-weak competition, the solutions converge uniformly to a positive equilibrium, enabling the coexistence of both Aedes aegypti and Aedes albopictus . We conduct numerical simulations on the two-species competition model along Route 441 in Florida, aiming to illustrate the invasion dynamics and competitive interactions of Aedes aegypti and Aedes albopictus .

Competitive exclusion principle, which states that two or more species limited by the same resource cannot coexist indefinitely, is a very common phenomenon in population dynamics. It is well-known that competitive exclusion principle occurs in deterministic competition models, diffusive competition models, and evolutionary competition models. In this paper, we consider an age-structured competition model among N species and obtain an interesting result: under suitable scaled birth and death rates, the species with the smallest maximum age always wins the competition to exclude the other species; that is, the competitive exclusion principle occurs in age-structured competition models.

Pierre Magal (1967–2024) was a leading researcher at the interface of differential equations, functional analysis, and mathematical biology. He made substantial contributions to both theoretical and applied aspects of these subjects. He published a dozen monographs, proceedings, and special issues and more than 125 journal articles. In this article we provide an introduction to Pierre’s contributions in some topics, including discrete population dynamics, integrated semigroup theory and abstract Cauchy problems with nondense domain, traveling waves in biological systems, uniform persistence and global attractors, cell-to-cell P-glycoprotein transfer in breast cancers, transfer problems in population dynamics and economics, and modeling of various epidemic problems, in particular his recent and extensive work on modeling COVID-19.