Yuhua Cai

We investigate the coevolutionary dynamics in a predator–prey system with Holling-III functional response. By using the theory of adaptive dynamics, we first classify coevolutionary singular coalitions and then prove the existence of cyclic coevolution. Coupling this cyclic trait evolution with oscillatory population dynamics generates multi-timescale oscillations in population size. These oscillations arise from the interplay of fast ecological and slow evolutionary processes, producing distinct patterns such as frequency-modulated and bursting oscillations. Our results demonstrate that complex oscillations in population size emerge intrinsically from eco-evolutionary feedback loops.

In this paper, we study the bifurcations (the birth and the annihilation) of relaxation oscillations in a nonsmooth, slow–fast biological model with predator mutual interference. We find that the number of relaxation oscillations will change via the bifurcation events including canard explosion and two homoclinic bifurcations under variations of the control parameter. During this dynamical process, the number of positive equilibria changes and their positions on the critical curve move. Based on geometric singular perturbation theory, canard theory and entry–exit function, we capture this dynamical process completely in a continuous way. Depending on the values of parameters, the number of relaxation oscillations in this model may be zero, one and at most two. All the theoretical predictions are verified by numerical simulations.

This paper studies the bifurcation mechanisms and the number of relaxation oscillations in a singular perturbation Leslie-Gower model with Allee effect in prey and cubic type piecewise nonlinear functional response. Our approaches are based on slow-fast normal form, entry-exit function and geometric singular perturbation theory. Under two topologically different assumptions on the critical curve, we find that the organization centers on the generation of relaxation oscillations are significantly different. When the left fold is lower than the right one, relaxation oscillations are created or vanish via supercritical and subcritical canard explosions and homoclinic bifurcation. The maximal number of relaxation oscillations in this scenario is three. However, when the left fold is higher than the right one, relaxation oscillations are created or vanish via inverse canard explosions and super-explosion. The maximal number of relaxation oscillations in this scenario is only one left since the transcritical bifurcation point can not enter into the generation of relaxation oscillations after the unique equilibrium passing through the left fold. All the theoretical predictions are verified by numerical simulations.

This article is concerned with the existence and spectral stability of pulses in singularly perturbed two-component reaction-diffusion systems with slowly mixed nonlinearity. In this paper, the slow nonlinearity is referred to be “mixed” in the sense that it is generated by a trigonometric function multiplied by a power function. We demonstrate via geometric singular perturbation theory that this model can support both the single-pulse and the double-hump solutions. The presence of the slowly mixed nonlinearity complicates the stability analysis on pulses, since the conditions that govern their stability can no longer be explicitly computed. We remove this difficulty by introducing the hypergeometric functions followed by a comparison theorem. By doing so, the “slow–fast” eigenvalues can be determined via the nonlocal eigenvalue problem method. We prove that the double-hump solution is always unstable, while the single-pulse solution can be stable under certain parameter conditions.

We study resident-invader dynamics in fluctuating environments when the invader and the resident have close but distinct strategies. First we focus on a class of continuous-time models of unstructured populations of multi-dimensional strategies, which incorporates environmental feedback and environmental stochasticity. Then we generalize our results to a class of structured population models. We classify the generic population dynamical outcomes of an invasion event when the resident population in a given environment is non-growing on the long-run and stochastically persistent. Our approach is based on the series expansion of a model with respect to the small strategy difference, and on the analysis of a stochastic fast-slow system induced by time-scale separation. Theoretical and numerical analyses show that the total size of the resident and invader population varies stochastically and dramatically in time, while the relative size of the invader population changes slowly and asymptotically in time. Thereby the classification is based on the asymptotic behavior of the relative population size, and which is shown to be fully determined by invasion criteria (i.e., without having to study the full generic dynamical system). Our results extend and generalize previous results for a stable resident equilibrium (particularly, Geritz in J Math Biol 50(1):67–82, 2005; Dercole and Geritz in J Theor Biol 394:231-254, 2016) to non-equilibrium resident population dynamics as well as resident dynamics with stochastic (or deterministic) drivers.

To understand the choice and competition of sites in nature, we consider an ecological environment in a chemostat consisting of a polymorphic microbial population that can float in the fluid or settle down on the wall of the chemostat. For the transition of a microbe from its floating state to its settled state at a particular settling rate involving the choice and competition of sites on the wall, we consider three different mechanisms: ( i ) unimolecular-Bourgeois settling , i.e., floaters land freely on the wall, but in an occupied site, the owner keeps the site (Bourgeois behaviour); ( ii ) unimolecular-anti-Bourgeois settling , i.e., floaters land freely on the wall, but in an occupied site, the intruder gets the site (anti-Bourgeois behaviour); ( iii ) bimolecular settling , i.e., floaters land only on the vacant sites of the wall. Employing the framework of adaptive dynamics, we study the evolution of the settling rate with different settling mechanisms and investigate how physical operating conditions affect the evolutionary dynamics. Our results indicate that settling mechanisms and physical operating conditions have significant influences on the direction of evolution and the diversity of phenotypes. (1) For constant nutrient input, theoretical analysis shows that the population is always monomorphic during the long-term evolution. Notably, the direction of evolution depends on the settling mechanisms and physical operating conditions, which further determines the composition of the population. Moreover, we find two exciting transformations of types of Pairwise Invasibility Plots, which are the gradual transformation and the bang-bang transformation . (2) For periodic nutrient input, numerical analysis reveals that evolutionary coexistence is possible, and the population eventually maintains dimorphism. Remarkably, for all three settling mechanisms, the long-term evolution leads to one of the two coexisting species settle down totally on the wall if the input is low-frequency but float entirely in the fluid if the input becomes high-frequency.