King-Yeung Lam

We review recent developments in the study of spreading phenomena in reaction–diffusion equations arising from ecological invasion models. Motivated by the conjecture of Shigesada and Kawasaki on staged invasions, we discuss how competition systems can lead to effective scalar models with shifting habitats. We present the Hamilton–Jacobi approach for determining spreading speeds and revisit the result of Li–Bewick–Shang–Fagan (2014) from this perspective. We then describe the emergence of nonlocally pulled fronts when the shifting habitat connects regions of distinct positive growth rates. Recent results including the works of Lam–Yu (2022) and Lam–Nadin–Yu (2025) are surveyed. We also discuss spreading phenomena in competition systems, predator-prey systems, and the existence of various classes of entire solutions. Finally, we discuss the logarithmic correction for invasion waves in moving environments and prove a new result.

We study the mechanism and effects of ideal free distributions (IFDs) on an ecological community consisting of n prey and m predator species, for any positive integers n and m , by considering the corresponding diffusive Lotka-Volterra system with time-periodic coefficients. We define a notion of joint IFD in a timeperiodic environment, and give necessary and sufficient conditions for it to be achieved by a subcollection of prey and predator species with suitable dispersal strategies. Next, we show, via construction of a Lyapunov function, that such dispersal strategies are evolutionarily stable, in the sense that if a subcollection of prey and predator species adopts an ideal free dispersal strategy, then the total community must converge to an IFD for large time; if a unique combination of prey-predator species adopts an ideal free strategy, then it can drive all other species to extinction. Conversely, if a combination of prey-predator species adopts a non-ideal free dispersal strategy, then it can be invaded by some suitable mutant strategies. Our results provide insight into the evolution of spatial distribution of ecological communities with predator-prey interactions.

Bird migration is an adaptive behavior that ultimately aims to optimize survival and reproductive success. We propose an optimal switching model in which migration behaviors are formulated as a stochastic optimal switching problem. In addition to considering the effect of stopover-site quality, we also account for the effect of information regarding conditions at the destination prior to arrival. For individuals with perfect information, we use numerical methods to evaluate the expected payoff and the corresponding optimal control. For individuals with partial information, we combine deterministic and stochastic simulations to investigate the associated strategy.

This paper is concerned with the following stationary Shigesada–Kawasaki–Teramoto competition system with cross-diffusion [Formula: see text] where [Formula: see text] and [Formula: see text] represent the densities of two competing species, [Formula: see text] is a bounded domain in [Formula: see text], and [Formula: see text] denotes the outer unit normal to [Formula: see text]. All coefficients [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] are assumed to be positive constants. The existence and stability/instability of nonconstant positive solutions of the above system have been widely studied in the literature but confined to large [Formula: see text] and small [Formula: see text] (or [Formula: see text] close to some particular number) with [Formula: see text]. In this paper, we establish the existence/nonexistence of nonconstant positive solutions for any [Formula: see text] and large [Formula: see text], which fills some gaps left out in the existing results. First, we show there are no positive solutions in the case of [Formula: see text] for large [Formula: see text]. Then by studying the shadow system of the above system as [Formula: see text], we establish the existence of positive solutions for large [Formula: see text] in various ranges of [Formula: see text] including all possible competitions: weak, strong-weak, and strong. In particular, we find some conditions under which multiple positive solutions exist. Finally we show the existence of positive solutions for some [Formula: see text] in the case of weak competition [Formula: see text].

The ideal free distribution in ecology was introduced by Fretwell and Lucas to model the habitat selection of animal populations. In this paper, we revisit the concept via a mean field game system with local coupling, which models a dynamic version of the habitat selection game in ecology. We establish the existence of classical solution of the ergodic mean field game system, including the case of heterogeneous diffusion when the underlying domain is one-dimensional and further show that the population density of agents converges to the ideal free distribution of the underlying habitat selection game, as the cost of control tends to zero. Our analysis provides a derivation of ideal free distribution in a dynamical context.

In this paper, we investigate the location of the spreading front and convergence to traveling wave profile of solutions to the Fisher-KPP equation in the following two cases: (i) in unbounded domains with an expanding boundary; (ii) on the real line where the environment function has a shifting jump discontinuity. Our approach is based on extending ideas in Bramson's seminal work in 1983, and applying gluing technique to construct super/subsolutions.