KingYeung Lam (Adrian)
Site Navigation
88th Midwest PDE Seminar
Location: Ohio State University,
Columbus, OH
Date: April 2628, 2024
Coorganize with Bo Guan, John Holmes, Yulong Xing
Registration/Workshop URL
BIRS Workshop: Dynamical Models Inspired by Biology (24w5209)
Location: Banff International Research Station
Banff National Park, Canada
Date: October 611, 2024
Coorganize with Avner Friedman, Yuan Lou, Bo Zhang, Bei Hu and Vincent Calvez
IMSI Workshop: Mathematical Modeling of Biological Interfacial Phenomena
Location: IMSI, University of Chicago
Chicago, IL
Date: Dec 913, 2024
Coorganize with Bei Hu and Benoit Perthame
Short course at the 2022 Spring Trimester Program at
Institut Henri Poinaré: Mathematical modeling of organization in living matter
Link to the program
Link to the course materials
Submitted Articles/Preprints:
K.Y. Lam, X.Q. Zhao and M. Zhu, Global dynamics of reactiondiffusion systems with a timevarying domain,
Submitted, 22pp.

Research Monograph:
1 
K.Y. Lam and Y. Lou,
Introduction to ReactionDiffusion Equations: Theory and Applications to Spatial Ecology and Evolutionary Biology, in press with Springer Cham, December 2022, 324pp.
(Link to Springer.com)
Table of Contents[Part I Linear Theory] 1. The maximum principle and principal eigenvalue for single equations; 2. The principal eigenvalue for periodicparabolic problems ; 3. The maximum principle and principal eigenvalue for systems; 4. The principal Floquet bundle for parabolice quations; 
Chap.1 
Book Chapter / Survey Articles:
3 
K.Y. Lam, Getting Your Hands Dirty: Teaching Math Biology with Active Learning Strategies. Early Career Section,
Bull. Amer. Math. Soc., Vol. 41, No. 4, April 2024, pp. 1012.
(DOI


2 
K.Y. Lam, S. Liu, and Y. Lou, Selected topics on reactiondiffusionadvection models from spatial ecology.
Math. Appl. Sci. Eng., Vol. 1 No. 2 (2020): pp. 91206.
(DOI 10.5206/mase/10644).
Read abstractWe discuss the effects of movement and spatial heterogeneity on population dynamics via reaction–diffusionadvection models, focusing on the persistence, competition, and evolution of organisms in spatially heterogeneous environments. Topics include LoktaVolterra competition models, river models, evolution of biased movement, phytoplanktongrowth, and spatial spread of epidemic disease. Open problems and conjectures are presented.P arts of this survey are adopted from the materials in [89,108,109], and some very recent progress are also included. 

1 
K.Y. Lam and Y. Lou,
Persistence, Competition, and Evolution.
In: Bianchi A., Hillen T., Lewis M., Yi Y. (eds) The Dynamics of Biological Systems.
Mathematics of Planet Earth, vol 4. Springer, Cham (2020) 31pp. (DOI 10.1007/9783030225834_8). Read abstractIn this chapter we discuss some reactiondiffusion models for the persistence of a single species, the competition of two populations, and the evolution of dispersal in spatially heterogeneous environments and advective environments. Our goal is to illustrate some interesting, and perhaps surprising, effects of spatial heterogeneity and diffusion on the population dynamics. Specific topics include the logistic model, linear eigenvalue problem with indefinite weight, LotkaVolterra competition models, reactiondiffusion models in advective environments, and the evolution of dispersal.chrome://bookmarks/ We will introduce some basic tools for reactiondiffusion equations such as the supersub solution method, the variational principle for principle eigenvalues, Lyapunov functionals, comparison principles for parabolic equations and systems, etc. Some recent developments will be discussed. In addition, problems with various difficulties ranging from elementary exercises to open research questions will be presented. 
PeerReviewed Publications:
53 
K.Y. Lam, Y. Lou and S. Ma, Exploring the evolutionary dynamics of infectious diseases through SIS
epidemic models,
Communications in Information and Systems (CIS) special issue dedicated to Prof. Avner Friedman’s 90th birthday,
accepted, 25pp.


52 
K.Y. Lam and Y. Lou,
The principal Floquet bundle and the dynamics of fast diffusing communities, Tran. Amer. Math. Soc., 377 (2024), 129.
(DOI: 10.1090/tran/8975).
Read abstractFor $N \geq 2$, consider the system of $N$ competing species which are ecologically identical and having distinct diffusion rates $\{D_i\}_{i=1}^N$, in an environment with the carrying capacity $m(x,t)$. For a generic class of $m(x,t)$ that varies with space and time, we show that there is a positive number $D_*$ independent of $N$ so that if $D_i \geq D_*$ for all {$1\le i\le N$}, then the slowest diffusing species is able to competitively exclude the rest of the species. In the case when the environment is temporally constant or temporally periodic, our result provides some further evidence in the affirmative direction regarding the conjecture by Dockery et al. in 1998. The main tool is the theory of the principal Floquet bundle for linear parabolic equations. 

51 
D. Gomez, K.Y. Lam and Y. Mori,
Front Propagation in the Shadow WavePinning Model, J. Math. Biol.,
86, 72 (2023).
(DOI: 10.1007/s00285023019086).
Read abstractIn this paper we consider a nonlocal bistable reactiondiffusion equation which is a simplified version of the wavepinning model of cell polarization. In the small diffusion limit, a typical solution $u(x,t)$ of this model approaches one of the stable states of the bistable nonlinearity in different parts of the spatial domain $\Omega$, separated by an interface moving at a normal velocity regulated by the integral $\int_\Omega u(x,t)\,dx$. In what is often referred to as wavepinning, feedback between massconservation and bistablity causes the interface to slow and approach a fixed limit. In the limit of a small diffusivity $\ep^2\ll 1$, we prove that for any $0<\gamma<1/2$ the interface can be estimated within $O(\ep^\gamma)$ of the location as predicted using formal asymptotics. We also discuss the sharpness of our result by comparing the formal asymptotic results with numerical simulations. 

50 
C. Heggerud, K.Y. Lam and H. Wang,
Niche differentiation in the light spectrum promotes coexistence of phytoplankton species: a spatial modelling approach, J. Math. Biol. 86 (2023), 54.
(With open access)
(DOI 10.1007/s0028502301890z).
Read abstractThe paradox of the plankton highlights the apparent contradiction between Gause’s law of competitive exclusion and the observed diversity of phytoplankton. It is well known that phytoplankton dynamics depend heavily on light availability. Here we treat light as a continuum of resources rather than a single resource by considering the visible light spectrum. We propose a spatially explicit reaction–diffusion–advection model to explore under what circumstance coexistence is possible from mathematical and biological perspectives. Furthermore, we provide biological context as to when coexistence is expected based on the degree of niche differentiation within the light spectrum and overall turbidity of the water. 

49 
K.Y. Lam, R. Lee and Y. Lou,
Population Dynamics in an Advective Environment, Comm. Appl. Math. Comp., in press.
(DOI 10.1007/s42967023002599).
Read abstractWe consider a onedimensional reaction–diffusion equation describing single and twospecies population dynamics in an advective environment, based on the modeling frameworks proposed by Lutscher, Lewis, and McCauley in 2006. We analyze the effect of rate of loss of individuals at both the upstream and downstream boundaries. In the singlespecies case, we prove the existence of the critical domain size and provide explicit formulas in terms of model parameters. We further derive qualitative properties of the critical domain size and show that, in some cases, the critical domain size is either strictly decreasing over all diffusion rates, or monotonically increasing after first decreasing to a minimum. We also consider competition between species differing only in their diffusion rates. For two species having large diffusion rates, we give a sufficient condition to determine whether the faster or slower diffuser wins the competition. We also briefly discuss applications of these results to competition in species whose spatial niche is affected by shifting isotherms caused by climate change. 

48 
K.Y. Lam, Y. Lou and B. Perthame,
A HamiltonJacobi Approach to Evolution of Dispersal, Comm. Partial Differential Equations, 48(1) (2022) 86118. (With open access)
(DOI 10.1080/03605302.2022.2139723).
Read abstractThe evolution of dispersal is a classical question in evolutionary biology, and it has been studied in a wide range of mathematical models. A selectionmutation model, in which the population is structured by space and a phenotypic trait, with the trait acting directly on the dispersal (diffusion) rate, was formulated by Perthame and Souganidis [Math. Model. Nat. Phenom. 11:154–166, 2016] to study the evolution of random dispersal toward the evolutionarily stable strategy. For the rare mutation limit, it was shown that the equilibrium population concentrates on a single trait associated to the smallest dispersal rate. In this paper, we consider the corresponding evolution equation and characterize the asymptotic behaviors of the timedependent solutions in the rare mutation limit, under mild convexity assumptions on the underlying Hamiltonian function. 

47 
K.Y. Lam and X. Yu,
Asymptotic spreading of KPP reactive fronts in heterogeneous shifting environments, J. Math. Pures Appl., 167 (2022) 147.
(DOI 10.1016/j.matpur.2022.09.001).
Read abstractWe study the asymptotic spreading of KolmogorovPetrovskyPiskunov (KPP) fronts in heterogeneous shifting habitats, with any number of shifting speeds, by further developing the method based on the theory of viscosity solutions of HamiltonJacobi equations. Our framework addresses both reactiondiffusion equation and integrodifferential equations with a distributed timedelay. The latter leads to a class of limiting equations of HamiltonJacobitype depending on the variable $x/t$ and in which the time and space derivatives are coupled together. We will first establish uniqueness results for these HamiltonJacobi equations using elementary arguments, and then characterize the spreading speed in terms of a reduced equation on a onedimensional domain in the variable $s=x/t$. In terms of the standard FisherKPP equation, our results leads to a new class of ``asymptotically homogeneous" environments which share the same spreading speed with the corresponding homogeneous environments. 

46 
A. Friedman, W. Hao and K.Y. Lam,
A cancer model with nonlocal free boundary dynamics, J. Math. Biol. Vol. 85, Article number: 46 (2022), 28pp.
(DOI 10.1007/s00285022018134).
Read abstractCancer cells at the tumor boundary move in the direction of \lam{the} oxygen gradient, while cancer cells far within the tumor are in a necrotic state. This paper introduces a simple mathematical model that accounts for these facts. The model consists of cancer cells, cytotoxic T cells, and oxygen satisfying a system of partial differential equations. Some of the model parameters represent the effect of anticancer drugs. The tumor boundary is a free boundary whose dynamics is determined by the movement of cancer cells at the boundary. The model is simulated for radially symmetric and axially symmetric tumors, and it is shown that the tumor may increase or decrease \lam{in size,} depending on the ``strength" of the drugs. Existence theorems are proved, global intime in the radially symmetric case, and local intime for any shape of tumor. In the radially symmetric case, it is proved, under different conditions, that the tumor may shrink monotonically, or expand monotonically. 

45 
B. Zhang, K.Y. Lam, W.M. Ni, K. M. Collins, Z. Fu, L. Zhai, Y. Lou, D.L. DeAngelis and A. Hastings, Directed movement changes coexistence outcomes in heterogeneous environments, Ecol. Lett., 25 (2022) 366377.
(DOI 10.1111/ele.13925).
Read abstractUnderstanding the mechanisms that promote coexistence is a central topic in ecology. Among plausible mechanisms of coexistence are different dispersal rates of competing types in heterogeneous environments. A large body of mathematical theory, based on LotkaVolterralike equations, proved noncoexistence, with the slower dispersing competitor always excluding the faster disperser. Notably, these results have not been tested empirically and these predictions considered mostly symmetric dispersal. To address these gaps, we not only extended previous theory by including exploitable resources and a component of directed movement, but also tested the new mathematical results experimentally using laboratory populations of the nematode worm, Caenorhaditis elegans. In contrast to previous predictions, our novel results reveal, both theoretically and empirically, that even with different symmetric dispersal, two identical competing species with moderate levels of directed components can coexist. Our results advance coexistence theories and underscore the role of directed dispersal in species coexistence in heterogeneous environments. 

44 
S. Liu, Q. Liu and K.Y. Lam,
Asymptotic spreading of interacting species with multiple fronts II:
Exponentially decaying initial data, J. Differential Equations, 303 (2021) 407455.
(DOI 10.1016/j.jde.2021.09.023).
Read abstractThis is part two of our study on the spreading properties of the weak LotkaVolterra competitiondiffusion systems with a stable coexistence state. Here we focus on the case when the initial data are exponential decaying. Here we establish a comparison principle for HamiltonJacobi equations, which enable us to adopt the HamiltonJacobi approach for FisherKPP equation due to Freidlin, Evans and Souganidis. As a result, the exact formulas of spreading speeds and their dependence on initial data are derived. Our results indicate that sometimes the spreading speed of the slower species is nonlocally determined. Connections of our results with the traveling profile due to Tang and Fife, as well as the more recent spreading result of Girardin and Lam, will be discussed. 

43 
R.S. Cantrell and K.Y. Lam,
On the evolution of slow dispersal in multispecies
communities, SIAM Math. Anal., 53(4), 4933–4964. (32 pages)
(DOI 10.1137/20M1361419).
Read abstractFor any $N \geq 2$, we show that there are choices of diffusion rates $\{d_i\}_{i=1}^N$ such that for $N$ competing species which are ecologically identical and having distinct diffusion rates, the slowest diffuser is able to competitive exclude the remainder of the species. In fact, the choices of such diffusion rates is open in the Hausdorff topology. Our result provdes some evidence in the affirmative direction regarding the conjecture in [Dockery et al., J. Math. Biol. (1998)]. The main tools include Morse decomposition of the semiflow, as well as the theory of normalized principal bundle for linear parabolic equations. A critical step in the proof is to establish the smooth dependence of the Floquet bundle on diffusion rate and other coefficients, which may be of independent interest. 

42 
H. Jiang, K.Y. Lam and Y. Lou,
Threepatch models for the evolution of dispersal in advective environments: varying drift and network topology, Bull. Math. Biol., 83, 109 (2021).
(DOI 10.1007/s11538021009398)Read abstractWe study the evolution of dispersal in advective threepatch models with distinct network topologies. Organisms can move between connected patches freely and they are also subject to the passive, directed drift. The carrying capacity is assumed to be the same in all patches, while the drift rates could vary. We first show that if all drift rates are the same, the faster dispersal rate is selected for all three models. For general drift rates, we show that the infinite diffusion rate is a local Convergence Stable Strategy (CvSS) for all three models. However, there are notable differences for three models: For Model I, the faster dispersal is always favored, irrespective of the drift rates, and thus the infinity dispersal rate is a global CvSS. In contrast, for Models II and III, a singular strategy will exist for some ranges of drift rates and bistability phenomenon happens, i.e. both infinity and zero diffusion rates are local CvSSs. Furthermore, for both Models II and III, it is possible for two competing populations to coexist by varying drift and diffusion rates. Some predictions on the dynamics of $n$patch models in advective environments are given. 

41 
A. Friedman and K.Y. Lam,
Analysis of a mathematical model of immune response to fungal infection, J. Math. Biol., 83, 8 (2021).
(DOI 10.1007/s0028502101633y).
Read abstractFungi are cells found as commensal residents on the skin and on mucosal surfaces of the human body, including the digestive track and urogenital track, but some species are pathogenic. Fungal infection may spread into deepseated organs causing lifethreatening infection, especially in immunecompromised individuals. Effective defense against fungal infection requires a coordinated response by the innate and adaptive immune systems. In the present paper we introduce a simple model of immune response to fungal infections consisting of three partial differential equations, for the populations of fungi ($F$), neutrophils ($N$) and cytotoxic T cells ($T$), taking $N$ and $T$ to represent, respectively, the innate and adaptive immune cells. We denote by $\lambda_F$ the aggressive proliferation rate of the fungi, by $\eta$ and $\zeta$ the killing rates of fungi by neutrophils and T cells, and by $N_0$ and $T_0$ the immune strengths, respectively, of $N$ and $T$ of an individual. Then take the expression $I=\eta N_0 + \zeta T_0  \lambda_F$ to represent the coordinated defense of the immune system against fungal infection. We use mathematical analysis to prove the following: If $I>0$, then the infection is eventually stopped and $F \to 0$ as $t \to \infty$; and (ii) if $I<0$ then the infection cannot be stopped and $F$ converges to some positive constant as $t\to\infty$. Treatments of fungal infection include antifungal agents and immunotherapy drugs, and both cause the parameter $I$ to increase. This is a postpeerreview, precopyedit version of an article published in [insert journal title]. The final authenticated version is available online at: http://dx.doi.org/10.1007/s0028502101633y”. 

40 
D. Jiang, Y. Lou and K.Y. Lam,
Competitive exclusion in a nonlocal reactiondiffusionadvection model of
phytoplankton populations,
Nonlinear Anal. Real World Appl., 61 (2021) 103350, 15pp.
(DOI 10.1016/j.nonrwa.2021.103350).
Read abstractWe continue our study on the global dynamics of a nonlocal reactiondiffusionadvection system modeling the population dynamics of two competing phytoplankton species in an eutrophic environment, where both populations depend solely on light for their metabolism. In our previous work \cite{Jiang2019}, we proved that system \eqref{equ31} is a strongly monotone dynamical system with respect to a nonstandard cone related to the cumulative distribution functions, and further determined the global dynamics when the species have either identical diffusion rate or identical advection rate. In this paper, we study the tradeoff of diffusion and advection and their joint influence on the outcome of competition. Two critical curves for the local stability of two semitrivial equilibria are analyzed, and some new competitive exclusion results are obtained. Our main tools, besides the theory of monotone dynamical system, include some new monotonicity results for the principal eigenvalues of elliptic operators in onedimensional domains. 

39 
R.S. Cantrell, C. Cosner and K.Y. Lam,
Ideal Free Dispersal under General Spatial heterogeneity and Time Periodicity, SIAM J. Appl. Math.,
81 (2021) 789813.
(DOI 10.1137/20M1332712).
Read abstractA population is said to have an ideal free distribution in a spatially heterogeneous but temporally constant environment if each of its members have chosen a fixed spatial location in a way that optimizes its individual fitness, allowing for the effects of crowding. In this paper, we extend the idea of individual fitness associated with a specific location in space to account for the full path that an individual organism takes in space and time over a periodic cycle, and extend the mathematical formulation of an ideal free distribution to general time periodic environments. We find that, as in many other cases, populations using dispersal strategies that can produce a generalized ideal free distribution have a competitive advantage relative to populations using strategies that do not produce an ideal free distribution. A sharp criterion on the environmental functions is found to be necessary and sufficient for such ideal free distribution to be feasible. In the case the criterion is met, we showed that there exist dispersal strategies that can be identified as producing a timeperiodic version of an ideal free distribution, and such strategies are evolutionarily steady and are neighborhood invaders from the viewpoint of adaptive dynamics. Our results extend previous works in which the environments are either temporally constant, or temporally periodic but the total carrying capacity is temporally constant 

38 
Q. Liu, S. Liu and K.Y. Lam Stacked invasion waves in a competitiondiffusion model with three species, J. Differential Equations, 271 (2021) 665718. (DOI 10.1016/j.jde.2020.09.008).
(The slightly longer version, containing the explicit formulas for the nonlocally pulled speed, can be found on arXiv:1910.04217)
Read abstractWe investigate the spreading properties of a threespecies competitiondiffusion system, which is noncooperative. We apply the HamiltonJacobi approach, due to Freidlin, Evans and Souganidis, to establish upper and lower estimates of spreading speed of the slowest species, in terms of the spreading speed of two faster species, and show that the estimates are sharp in some situations. The spreading speed will first be characterized as the free boundary point of the viscosity solution for certain variational inequality cast in the space of speeds. Its exact formulas will then be derived by solving the variational inequality explicitly. To the best of our knowledge, this is the first theoretical result on threespecies competition system in unbounded domains. 

37 
R.S. Cantrell and K.Y. Lam,
Competitive exclusion in phytoplankton communities in a eutrophic water column, Discrete Contin.Dyn. Syst. Ser. B, 26 (2021), 17831795.
(DOI 10.3934/dcdsb.2020361).
Read abstractWe analyze a reactiondiffusion system modeling the competition of multiple phytoplankton species which are limited only by light. While the dynamics of a single species has been well studied, the dynamics of the twospecies model has only begun to be understood with the recent establishment of a comparison principle. In this paper, we show that the competition of $N$ similar phytoplankton species, for any number $N$, generically leads to competition exclusion. The main tool is the theory of normalized principal bundle for linear parabolic equations. 

36 
W. Hao, K.Y. Lam and Y. Lou, Ecological and Evolutionary Dynamics in Advective Environments: Critical Domain Size and Boundary Conditions, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 367400 (DOI 10.3934/dcdsb.2020283).
Read abstractWe consider the ecological and evolutionary dynamics of a reactiondiffusionadvection model for populations residing in a onedimensional advective homogeneous environment, with emphasis on the effects of boundary conditions and domain size. We assume that there is a net loss of populations at the downstream end with rate $b \geq 0$, while the noflux condition is imposed on the upstream end. For the single species model, it is shown that the critical patch size is a decreasing function of the dispersal rate when $b \leq 3/2$; whereas it first decreases and then increases when $b >3/2$. For the twospecies competition model, we show that the infinite dispersal rate is evolutionarily stable for $b < 3/2$ and, when dispersal rates of both species are large, the population with larger dispersal rate always displace the population with smaller rate. For certain specific population loss rate $b<3/2$, it is also shown that there can be up to three evolutionarily stable strategies. For $b>3/2$, it is proved that the infinite random dispersal rate is not evolutionarily stable, and that, for some specific $b>3/2$, a finite dispersal rate is evolutionarily stable. Furthermore, for the intermediate domain size, this dispersal rate is optimal in the sense that the species adopting this rate is able to displace its competitor with a similar but different rate. Finally, nine qualitatively different pairwise invasibility plots are obtained by varying the parameter $b$ and the domain size. 

35 
H. Jiang, K.Y. Lam and Y. Lou, Are twopatch models sufficient? The evolution of dispersal and topology of river network modules, Bull. Math. Biol., 82, Article number: 131 (2020) 34pp. (DOI 10.1007/s11538020008031)
Read abstractWe study the dynamics of two competing species in threepatch models and illustrate how the topology of directed river network modules may affect the evolution of dispersal. Each model assumes that patch 1 is at the upstream end, patch 3 is at the downstream end, but patch 2 could be upstream, or middle stream, or downstream, depending on the specific topology of the modules. We posit that individuals are subject to both unbiased dispersal between patches and passive drift from one patch to another, depending upon the connectivity of patches. When the drift rate is small, we show that for all models, the mutant species can invade when rare if and only if it is the slower disperser. However, when the drift rate is large, most models predict that the faster disperser wins, while some predict that there exists one evolutionarily singular strategy. The intermediate range of drift is much more complex: most models predict the existence of one singular strategy, but it may or may not be evolutionarily stable, again depending upon the topology of modules, while one model even predicts that for some intermediate drift rate, singular strategy does not exist and the faster disperser wins the competition. This is a postpeerreview, precopyedit version of an article published in [insert journal title]. The final authenticated version is available online at: http://dx.doi.org/10.1007/s11538020008031”. 

34 
V. Calvez and K.Y. Lam,
Uniqueness of the viscosity solution of a constrained HamiltonJacobi equation, Cal. Var. PDE, 59, Article number: 163 (2020) 22pp. (arXiv:1809.05317). (DOI 10.1007/s00526020018190).
Read abstractIn quantitative genetics, viscosity solutions of HamiltonJacobi equations appear naturally in the asymptotic limit of selectionmutation models when the population variance vanishes. They have to be solved together with an unknown function I(t) that arises as the counterpart of a nonnegativity constraint on the solution at each time. Although the uniqueness of viscosity solutions is known for many variants of HamiltonJacobi equations, the uniqueness for this particular type of constrained problem was not resolved, except in a few particular cases. Here, we provide a general answer to the uniqueness problem, based on three main assumptions: convexity of the Hamiltonian function H(I, x, p) with respect to p, monotonicity of H with respect to I, and BV regularity of I(t). 

33 
K.Y. Lam R. Salako, and Q. Wu,
Entire Solutions of Diffusive LotkaVolterra System,
J. Differential Equations, 269 (2020) 1075810791.
(10.1016/j.jde.2020.07.006).
Read abstractThis work is concerned with the existence of entire solutions of the diffusive LotkaVolterra competition system. We prove the existence of several new classes of entire solutions. Moreover, we also describe the largetime asymptotic behavior of these entire solutions. In the weak competition case, we prove the existence of a class of entire solutions that forms a 4dimensional manifold. 

32 
A. Friedman and K.Y. Lam,
Analysis of a mathematical model of rheumatoid arthritis,
J. Math. Biol., 80 (2020) 18571883.
(DOI 10.1007/s00285020014821).
Read abstractRheumatoid arthritis is an autoimmune disease characterized by inflammation in the synovial fluid within the synovial joint connecting two contiguous bony surfaces. The inflammation diffuses into the cartilage adjacent to each of the bony surfaces, resulting in their gradual destruction. The interface between the cartilage and the synovial fluid is an evolving free boundary. In this paper we consider a twophase free boundary problem based on a simplified model of rheumatoid arthritis. We prove global existence and uniqueness of a solution, and derive properties of the free boundary. In particular it is proved that the free boundary increases in time, and the cartilage shrinks to zero as t → ∞, even under treatment by a drug. It is also shown in the reduced onephased problem, with cartilage alone, that a larger prescribed inflammation function leads to a faster destruction of the cartilage. This is a postpeerreview, precopyedit version of an article published in [insert journal title]. The final authenticated version is available online at: http://dx.doi.org/10.1007/s00285020014821”. 

31 
Q. Liu, S. Liu and K.Y. Lam,
Asymptotic spreading of interacting species with multiple fronts I: A geometric optics approach, Discrete Cont. Dyn. Syst. Ser. A 40 (2020) 36833714.
(DOI 10.3934/dcds.2020050).
Read abstractWe establish spreading properties of the LotkaVolterra competitiondiffusion system. When the initial data vanish on a right halfline, we derive the exact spreading speeds and prove the convergence to homogeneous equilibrium states in between the interfaces. Our method is inspired by the geometric optics approach for FisherKPP equation due to Freidlin, Evans and Souganidis. Our main result settles an open question raised in a paper of Lin and Li (2012), and show that one of the species spreads to the right with a nonlocally determined speed. 

30 
L. Girardin and K.Y. Lam (2019),
Invasion of open space by two competitors: spreading properties of monostable twospecies competitiondiffusion systems,
Proc. Lond. Math. Soc., 119 (2019), 12791335.
(DOI 10.1112/plms.12270).
PLMS Top Downloaded Paper 20182019
Read abstractThis paper is concerned with some spreading properties of monostable LotkaVolterra twospecies competitiondiffusion systems when the initial values are null or exponentially decaying in a right halfline. Thanks to a careful construction of supersolutions and subsolutions, we improve previously known results and settle open questions. In particular, we show that if the weaker competitor is also the faster one, then it is able to evade the stronger and slower competitor by invading first into unoccupied territories. The pair of speeds depends on the initial values. If these are null in a right halfline, then the first speed is the KPP speed of the fastest competitor and the second speed is given by an exact formula describing the possibility of non local pulling. Furthermore, the unbounded set of pairs of speeds achievable with exponentially decaying initial values is characterized, up to a negligible set. 

29 
D. Jiang, Y. Lou K.Y. Lam and Z. Wang,
Monotonicity and global dynamics of a nonlocal twospecies phytoplankton model,
SIAM J. Appl. Math., 79 (2019), 716742.
(DOI 10.1137/18M1221588).
Read abstractWe investigate a nonlocal reactiondiffusionadvection system modeling the population dynamics of two competing phytoplankton species in a eutrophic environment, where nutrients are in abundance and the species are limited by light only for their metabolism. We first demonstrate that the system does not preserve the competitive order in the pointwise sense. Then we introduce a special cone K involving the cumulative distributions of the population densities, and a generalized notion of super and subsolutions of the nonlocal competition system where the differential inequalities hold in the sense of the cone K. A comparison principle is then established for such super and subsolutions, which implies the monotonicity of the underlying semiflow with respect to the cone K As application, we study the global dynamics of the single species system and the competition system. The latter has implications for the evolution of movement for phytoplankton species. 

28 
W. Hao, K.Y. Lam and Y. Lou,
Concentration phenomena in an integroPDE model for evolution of conditional dispersal,
Indiana Univ. Math. J., 68 (2019), 881923.
(DOI 10.1512/iumj.2019.68.7625).Read abstractTo study the evolution of conditional dispersal we extend the PerthameSouganidis mutationselection model and consider an integroPDE model for a population structured by the spatial variables and one trait vari able. We assume that both the diffusion rate and advection rate are functions of the trait variable, which lies within a short interval I. Competition for resource is local in spatial variables, but nonlocal in the trait variable. Under proper conditions on the invasion fitness gradient, we show that in the limit of small mutation rate, the positive steady state solution will concentrate in the trait variable and forms (i) a Dirac mass supported at one end of I; or (ii) a Dirac mass supported at the interior of I; or (iii) two Dirac masses supported at both ends of I, respectively. While Cases (i) and (ii) imply the evolutionary stability of a single strategy, Case (iii) suggests that when no single strategy can be evolutionarily stable, it is possible that two peculiar strategies as a pair can be evolutionarily stable and resist the invasion of any other strategy in our context. 

27 
L. Su, K.Y. Lam and R. Bürger,
Twolocus clines maintained by diffusion and recombination in a heterogeneous environment,
J. Differential Equations, 266 (2019), 79097947.
(10.1016/j.jde.2018.12.022).
Read abstractWe study existence and stability of stationary solutions of a system of semilinear parabolic partial differential equations that occurs in population genetics. It describes the evolution of gamete frequencies in a geographically structured population of migrating individuals in a bounded habitat. Fitness of individuals is determined additively by two recombining, diallelic genetic loci that are subject to spatially varying selection. Migration is modeled by diffusion. Of most interest are spatially nonconstant stationary solutions, socalled clines. In a twolocus cline all four gametes are present in the population, i.e., it is an internal stationary solution. We provide conditions for existence and linear stability of a twolocus cline if recombination is either sufficiently weak or sufficiently strong relative to selection and diffusion. For strong recombination, we also prove uniqueness and global asymptotic stability. For arbitrary recombination, we determine the stability properties of the monomorphic equilibria, which represent fixation of a single gamete. 

26 
X. He, K.Y. Lam, Y. Lou and W.M. Ni,
Dynamics of a ConsumerResource ReactionDiffusion Model,
J. Math. Biol., 78 (2019), 16051636.
(DOI 10.1007/s002850181321z).
Read abstractWe study the dynamics of a consumerresource reactiondiffusion model, proposed recently by Zhang et al (2017), in both homogeneous and heterogeneous environments. For homogeneous environments we establish the global stability of constant steady states. For heterogeneous environments we study the existence and stability of positive steady states and the persistence of timedependent solutions. Our results illustrate that for heterogeneous environments there are some parameter regions in which the resources are only partially limited in space, a unique feature which does not occur in homogeneous environments. Such difference between homogeneous and heterogeneous environments seems to be closely connected with a recent finding by Zhang et al (2017), which says that in consumerresource models, homogeneously distributed resources could support higher population abundance than heterogeneously distributed resources. This is opposite to the prediction by Lou (2006) for logistictype models. For both small and high yield rates, we also show that when a consumer exists in a region with a heterogeneously distributed input of exploitable renewed limiting resources, the total population abundance at equilibrium can reach a greater abundance when it diffuses than when it does not. In contrast, such phenomenon may fail for intermediate yield rates. This is a postpeerreview, precopyedit version of an article published in [insert journal title]. The final authenticated version is available online at: http://dx.doi.org/10.1007/s002850181321z”. 

25 
K.Y. Lam,
Diracconcentrations in an Integropde Model from Evolutionary Game Theory,
Discrete Cont. Dyn. Syst. Ser. B, 24 (2019), 737754. (DOI 10.3934/dcdsb.2018205).Read abstractNonlocal LotkaVolterra models have the property that solutions concentrate as Dirac masses in the limit of small diffusion. Motivated by the existence of moving Diracconcentrations in the timedependent problem, we study the qualitative properties of steady states in the limit of small diffusion. Under different conditions on the growth rate and interaction kernel as motivated by the framework of adaptive dynamics, we will show that as the diffusion rate tends to zero the steady state concentrates (i) at a single location; (ii) at two locations simultaneously; or (iii) at one of two alternative locations. The third result in particular shows that solutions need not be unique. This marks an important difference of the nonlocal equation with its local counterpart. 

24 
K.Y. Lam, X. Wang and T. Zhang,
Traveling waves for a class of diffusive diseasetransmission models with network structures,
SIAM J. Math. Anal., 50 (2018), 57195748.
(DOI 10.1137/17M1144258).Read abstractIn this paper, the necessary and sufficient conditions for the existence of traveling wave solutions are derived for a class of diffusive diseasetransmission models with network structures. The existence of traveling semifronts is obtained by Schauder's fixedpoint theorem, and these traveling semifronts are shown to be bounded by transforming the boundedness problem into the classification problem of nonnegative solutions to a linear elliptic system on R^1 . To overcome the reducibility problem arising in the proofs, Harnack's inequality for positive supersolutions on R^1 is proved. 

23 
R. S. Cantrell, C. Cosner and K.Y. Lam ,
On residentinvader dynamics in infinite dimensional dynamical systems,
J. Differential Equations, 263 (2017), 45654616. (51pp.)
(DOI 10.1016/j.jde.2017.05.029).Read abstractMotivated by evolutionary biology, we study general infinitedimensional dynamical systems involving two species  the resident and the invader. Sufficient conditions for competition exclusion phenomena are given when the two species play similar, but distinct, strategies. Those conditions are based on invasibility criteria, for instance, evolutionarily stable strategies in the framework of adaptive dynamics. These types of questions were first proposed and studied by [S. Geritz et al., J. Math. Biol., 2002] and [S. Geritz, J. Math. Biol., 2005] for a class of ordinary differential equa tions. We extend and generalize previous work in two directions. Firstly, we consider analytic semiflows in infinitedimensional spaces. Secondly, we devise an argument based on Hadamard’s graph transform method that does not depend on the monotonicity of the twospecies system. Our results are applicable to a wide class of reactiondiffusion models as well as models with nonlocal diffusion operators. 

22 
H.B. Hsu, K.Y. Lam and F.B. Wang,
Single species growth consuming inorganic carbon with internal storage in the unstirred chemostat,
J. Math. Biol., 75 (2017), 17751825. (51pp.)
(DOI 10.1007/s0028501711345).Read abstractThis paper presents a PDE system modeling the growth of a single species population consuming inorganic carbon that is stored internally in a poorly mixed habitat. Inorganic carbon takes the forms of “CO2” (dissolved CO2 and carbonic acid) and “CARB” (bicarbonate and carbonate ions), which are substitutable in their effects on algal growth. We first establish a threshold type result on the extinction/persistence of the species in terms of the sign of a principal eigenvalue associated with a nonlinear eigenvalue problem. If the habitat is the unstirred chemostat, we add biologically relevant assumptions on the uptake functions and prove the uniqueness and global attractivity of the positive steady state when the species persists. This is a postpeerreview, precopyedit version of an article published in [insert journal title]. The final authenticated version is available online at: http://dx.doi.org/10.1007/s0028501711345”. 

21 
K.Y. Lam,
Stability of Dirac Concentrations in an IntegroPDE Model for Evolution of Dispersal,
Cal. Var. PDE, (2017) 56: 79. (DOI 10.1007/s0052601711571).Read abstractWe consider an integroPDE model from evolutionary biology. The solution u(x,a) structured by two variables x and alpha. The diffusion coefficient in the x direction depends on a and the diffusion coefficient in the α direction is a constant epsilon. A special feature of this model is the appearance of the integral of the solution in the alpha variable, which can be viewed as an infinite dimensional parameter of the problem. In a previous work, the existence of a steady state that exhibits Diracconcentration in one of the variables yet remains regular in the other variables was proved independently by [Lam and Lou, J. Funct. Anal. (2017)] and [Perthame and Souganidies, arXiv:1505.03420 (2015)]. In this paper, we tackle the longtime dynamics of solutions. When the environment function is nonconstant, we show that the steady state is linearly stable by considering the corresponding nonlocal eigenvalue problem. Uniqueness of steady state is obtained from the stability result via a degree argument. When the environment function is constant, the global asymptotic stability result is obtained. This problem can be regarded as a competition of infinitely many species parameterized by α. As with the competition model for three or more species, the integroPDE model does not generate a monotone dynamical system so that it is necessary to consider all (real or complex) eigenvalues in determining its linear stability. 

20 
R. Cui, K.Y. Lam and Y. Lou,
Dynamics and Asymptotic Profiles of Steady States of an Epidemic Model in Advective Environments,
J. Differential Equations, 263 (2017), 23432373.
(10.1016/j.jde.2017.03.045) Read abstractWe study the dynamics of a SIS epidemic model of reactiondiffusionadvection type. The persistence of infected and susceptible populations and the global stability of the disease free equilibrium are established when the basic reproduction number is greater than or less than or equal to one, respectively. We further consider the effects of diffusion and advection on asymptotic profiles of endemic equilibrium: When the advection rate is relatively large comparing to the diffusion rates of both populations, then two populations persist and concentrate at the downstream end. As the diffusion rate of the susceptible population tends to zero, the density of the infected population decays exponentially for positive advection rate but linearly when there is no advection. Our results suggest that advection can help speed up the elimination of disease. 

19 
M. Golubitsky, W. Hao, K.Y. Lam and Y. Lou,
Dimorphism by singularity theory in a model for river ecology,
Bull. Math. Biol., 79 (2017), 10511069.
Read abstractGeritz, Gyllenberg, Jacobs, and Parvinen show that two similar species can coexist only if their strategies are in a sector of parameter space near a nondegenerate evolutionarily singular strategy. We show that the dimorphism region can be more general by using the unfolding theory of Wang and Golubitsky near a degenerate evolutionarily singular strategy. Specifically, we use a PDE model of river species as an example of this approach. Our find ing shows that the dimorphism region can exhibit various different forms that are strikingly different from previously known results in adaptive dynamics. This is a postpeerreview, precopyedit version of an article published in [insert journal title]. The final authenticated version is available online at: http://rdcu.be/J6M3”. 

18 
R.S. Cantrell, X. Cao, K.Y. Lam and T. Xiang,
A PDE model of intraguild predation with crossdiffusion,
Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 36533661.
(DOI 10.3934/dcdsb.2017145) Read abstractThis note concerns a quasilinear parabolic system modeling an intraguild predation community in a focal habitat in Rn, n greater than or equal to 2. In this system the intraguild prey employs a fitnessbased dispersal strategy whereby the in traguild prey moves away from a locale when predation risk is high enough to render the locale undesirable for resource acquisition. The system modifies the model considered in Ryan and Cantrell (2015) by adding an element of mutual interference among predators to the functional response terms in the model, thereby switching from Holling II forms to BeddingtonDeAngelis forms. We show that the resulting system can be realized as a semidynamical system with a global attractor for any n greater than or equal to 2. In contrast, the orginal model was restricted to two dimensional spatial habitats. The permanance of the intraguild prey then follows as in Ryan and Cantrell by means of the Acyclicity Theorem of Persistence Theory. 

17 
K.Y. Lam and Y. Lou,
An integroPDE model for evolution of random dispersal,
J. Funct. Anal., 272 (2017), 17551790.
(DOI 10.1016/j.jfa.2016.11.017)
Read abstractWe consider an integroPDE model for a population structured by the spatial variables and a trait variable which is the diffusion rate. Competition for resource is local in spatial variables, but nonlocal in the trait variable. We focus on the asymptotic profile of positive steady state solutions. Our result shows that in the limit of small mutation rate, the solution remains regular in the spatial variables and yet concentrates in the trait variable and forms a Dirac mass supported at the lowest diffusion rate. Hastings and Dockery et al. showed that for two competing species in spatially heterogeneous but temporally constant environment, the slower diffuser always prevails, if all other things are held equal. Our result suggests that their findings may well hold for arbitrarily many or even a continuum of traits. 

16 
I. Averill, K.Y. Lam and Y. Lou,
The role of advection in a twospecies competition model: a bifurcation approach,
Mem. Amer. Math. Soc., Vol. 245 (2017), no. 1161, v+117pp.
(DOI 10.1090/memo/1161) Read abstractThe effects of weak and strong advection on the dynamics of reaction diffusion models have long been studied. In contrast, the role of intermediate advec tion remains poorly understood. For example, concentration phenomena can occur when advection is strong, providing a mechanism for the coexistence of multiple populations, in contrast with the situation of weak advection where coexistence may not be possible. The transition of the dynamics from weak to strong advection is generally difficult to determine. In this work we consider a mathematical model of two competing populations in a spatially varying but temporally constant environment, where both species have the same population dynamics but different dispersal strategies: one species adopts random dispersal, while the dispersal strategy for the other species is a combination of random dispersal and advection upward along the resource gradient. For any given diffusion rates we consider the bifurcation diagram of positive steady states by using the advection rate as the bifurcation parameter. This approach enables us to capture the change of dynamics from weak advection to strong advection. We will determine three different types of bifurcation diagrams, depending on the difference of diffusion rates. Some exact multiplicity results about bifurcation points will also be presented. Our results can unify some previous work and, as a case study about the role of advection, also contribute to our understanding of intermediate (relative to diffusion) advection in reactiondiffusion models. 

15 
K.Y. Lam, Y. Lou and F. Lutscher,
The emergence of range limits in advective environments,
SIAM J. Appl. Math., 76 (2016), 641662.
(DOI 10.1137/15M1027887) Read abstractIn this paper, we study the asymptotic profile of the steady state of a reaction diffusionadvection model in ecology proposed in [E. Pachepsky et al., Theoret. Popul. Biol., 67 (2005), pp. 61–73; D. Speirs and W. Gurney, Ecology, 82 (2001), pp. 1219–1237]. The model de scribes the population dynamics of a single species experiencing a unidirectional flow. We show the existence of one or more internal transition layers and determine their locations. Such locations can be understood as the upstream invasion limits of the species. It turns out that these invasion limits are connected to the upstream spreading speed of the species and are sometimes subject to the effect of migration from upstream source patches. 

14 
K.Y. Lam and Y. Lou,
Asymptotic behavior of the principal eigenvalue for cooperative elliptic systems and applications,
J. Dynam. Differential Equations, 28 (2016), 2948.
(DOI 10.1007/s1088401595044)Read abstractThe asymptotic behavior of the principal eigenvalue for general linear cooperative elliptic systems with small diffusion rates is determined. As an application, we show that if a cooperative system of ordinary differential equations has a unique positive equilibrium which is globally asymptotically stable, then the corresponding reactiondiffusion system with either the Neumann boundary condition or the Robin boundary condition also has a unique positive steady state which is globally asymptotically stable, provided that the diffusion coefficients are sufficiently small. Moreover, as the diffusion coefficients approach zero, the positive steady state of the reactiondiffusion system converges uniformly to the equilibrium of the corresponding kinetic system. 

13 
K.Y. Lam and D. Munther,
A Remark on the Global Dynamics of Competitive Systems on Ordered Banach Spaces,
Proc. Amer. Math. Soc. 144 (2016), 11531159.
(DOI 10.1090/proc12768)
Read abstractA wellknown result in [HsuSmithWaltman, Trans. AMS (1996)] states that in a competitive semiflow defined on X+ = X1+ ×X2+, the product of two cones in respective Banach spaces, if (u∗,0) and (0,v∗) are the global attractors in X1+ × {0} and {0} × X2+ respectively, then one of the following three outcomes is possible for the two competitors: either there is at least one coexistence steady state, or one of (u∗,0),(0,v∗) attracts all trajectories initiating in the order interval I = [0, u∗] × [0, v∗]. However, it was demon strated by an example that in some cases neither (u∗,0) nor (0,v∗) is globally asymptotically stable if we broaden our scope to all of X+. In this paper, we give two sufficient conditions that guarantee, in the absence of coexistence steady states, the global asymptotic stability of one of (u∗, 0) or (0, v∗) among all trajectories in X+. Namely, one of (u∗,0) or (0,v∗) is (i) linearly unsta ble, or (ii) is linearly neutrally stable but zero is a simple eigenvalue. Our results complement the counter example mentioned in the above paper as well as applications that frequently arise in practice. 

12 
A. Friedman and K.Y. Lam,
Analysis of a freeboundary tumor model with angiogenesis,
J. Differential Equations 259 (2015), 76367661.
(DOI 10.1016/j.jde.2015.08.032) Read abstractWe consider a free boundary problem for a spherically symmetric tumor with free boundary. In order to receive nutrients u the tumor attracts blood vessel at a rate proportional to a given function alpha(t), so that a robin condition holds on the boundary depending on the nutrient concentration outside the tumor. A parameter in the model is proportional to the ‘aggressiveness’ of the tumor. When alpha is a constant, the existence and uniqueness of stationary solution is proved. For the more general situation when alpha depends on time, we show, under various conditions that (i) R(t) remains bounded if alpha(t) remains bounded; (ii) R(t) > 0 if alpha(t) >0; and (iii) liminf R(t) > 0 if liminf alpha(t) > 0. Surprisingly, we exhibit solutions where alpha(t) > 0 exponentially in t while R(t) > infinity exponentially in t. Finally, we prove the global asymptotic stability of steady state when μ is sufficiently small. 

11 
K.Y. Lam, Y. Lou and F. Lutscher ,
Evolution of dispersal in closed advective environments,
J. Biol. Dyn. 9 Suppl. 1 (2014), 188212.
(DOI 10.1080/17513758.2014.969336) Read abstractWe study a twospecies competition model in a closed advective environment, where individuals are exposed to unidirectional flow (advection) but no individuals are lost through the boundary. The two species have the same growth and advection rates but different random dispersal rates. The linear stability analysis of the semitrivial steady state suggests that, in contrast to the case without advection, slow dispersal is generally selected against in closed advective environments. We investigate the invasion exponent for various types of resource functions, and our analysis suggests that there might exist some intermediate dispersal rate that will be selected. When the diffusion and advection rates are small and comparable, we determine criteria for the existence and multiplicity of singular strategies and evolutionarily stable strategies. We further show that every singular strategy is convergent stable. 

10 
K.Y. Lam and W.M. Ni,
Advectionmediated competition in general environments,
J. Differential Equations, 257 (2014), 34663500.
(DOI 10.1016/j.jde.2014.06.019) Read abstractWe consider a reactiondiffusionadvection system of two competing species with one of the species dispersing by random diffusion as well as a biased movement upward along resource gradient, while the other species by random diffusion only. It has been shown that, under some nondegeneracy conditions on the environment function, the two species always coexist when the advection is strong. In this paper, we show that for general smooth environment function, in contrast to what is known, there can be competitive exclusion when the advection is strong, and, we give a sharp criterion for coexistence that includes all previously considered cases. Moreover, when the domain is onedimensional, we derive in the strong advection limit a system of two equations defined on different domains. Uniqueness of steady states of this nonstandard system is obtained when one of the diffusion rates is large. 

9 
A. Friedman and K.Y. Lam,
On the stability of steady states in a granuloma model,
J. Differential Equations, 256 (2014), 37433769.
(DOI 10.1016/j.jde.2014.02.019) Read abstractWe consider a free boundary problem for a system of two semilinear parabolic equations. The system represents a simple model of granuloma, a collection of immune cells and bacteria filling a 3 dimensional domain which varies in time. We prove the existence of stationary spherical solutions and study their linear asymptotic stability as time increases to infinity. 

8 
K.Y. Lam and D. Munther,
Invading the ideal free distribution, Special Issue in Honor of Chris Cosner's 60th Birthday ,
Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 32193244.
(DOI 10.3934/dcdsb.2014.19.3219) Read abstractRecently, the ideal free dispersal strategy has been proven to be evolutionarily stable in the spatially discrete as well as continuous setting. That is, at equilibrium a species adopting the strategy is immune against invasion by any species carrying a different dispersal strategy, other conditions being held equal. In this paper, we consider a twospecies competition model where one of the species adopts an ideal free dispersal strategy, but is penalized by a weak Allee effect. We will show rigorously in this case that the ideal free disperser is invasible by a range of nonideal free strategies, illustrating the tradeoff between the advantage of being an ideal free disperser and the setback caused by the weak Alee effect. Moreover, a sharp integral criterion is given to determine the stability/instability of one of the semitrivial steady state, which is always linearly neutrally stable due to the degeneracy caused by the weak Allee effect. 

7 
K.Y. Lam and Y. Lou,
Evolutionarily stable and convergent stable strategies in
reactiondiffusion models for conditional dispersal,
Bull. Math. Biol.,76 (2014), 261291.
(DOI 10.1007/s115380139901y)
Read abstractWe consider a mathematical model of two competing species for the evolution of conditional dispersal in a spatially varying but temporally constant environment. Two species are different only in their dispersal strategies, which are a combination of random dispersal and biased movement upward along the resource gradient. In the absence of biased movement or advection, Hastings showed that the mutant can invade when rare if and only if it has smaller random dispersal rate than the resident. When there is a small amount of biased movement or advection, we show that there is a positive random dispersal rate that is both locally evolutionarily stable and convergent stable. Our analysis of the model suggests that a balanced combination of random and biased movement might be a better habitat selection strategy for populations. This is a postpeerreview, precopyedit version of an article published in [insert journal title]. The final authenticated version is available online at: http://dx.doi.org/10.1007/s115380139901y”. 

6 
K.Y. Lam and Y. Lou,
Evolution of dispersal: ESS in spatial models,
J. Math. Biol., 68 (2014), 851877.
(DOI 10.1007/s0028501306501)
Read abstractWe consider a twospecies competition model in which the species have the same population dynamics but different dispersal strategies. Both species disperse by a combination of random diffusion and advection along environmental gradients, with the same random dispersal rates but different advection coefficients. Regarding these advection coefficients as movement strategies of the species, we investigate their course of evolution. By applying invasion analysis we find that if the spatial environmental variation is less than a critical value, there is a unique evolutionarily singular strategy, which is also evolutionarily stable. If the spatial environmental variation exceeds the critical value, there can be at least three evolutionarily singular strategies, one of which is not evolutionarily stable. Our results suggest that the evolution of conditional dispersal of organisms depends upon the spatial heterogeneity of the environment in a subtle way This is a postpeerreview, precopyedit version of an article published in [insert journal title]. The final authenticated version is available online at: http://dx.doi.org/10.1007/s0028501306501”. 

5 
K.Y. Lam,
Limiting Profiles of semilinear elliptic equations with large advection in population dynamics II,
SIAM J. Math. Anal., 44 (2012), 18081830.
(DOI 10.1137/100819758)
Read abstractLimiting profiles of solutions to a 2x2 Lotka–Volterra competitiondiffusionadvection system, when the strength of the advection tends to infinity, are determined. The two species, competing in a heterogeneous environment, are identical except for their dispersal strategies: one is just a random diffusion, while the other is “smarter”—a combination of random diffusion and a directed movement up the environmental gradient. In the previous paper of Lam and Ni [Discrete Contin. Dyn. Syst. 28 (2010), pp. 1051–1067], it is proved that in one space dimension, for large advection the “smarter” species concentrates near a selected subset of positive local maximum points of the environment function, establishing a conjecture proposed by Cantrell, Cosner, and Lou. With a different method, we generalize this result to any dimensions with the peaks located under mild hypotheses on the environment function. Moreover, a Liouvilletype result which gives the asymptotic profile is proved. 

4 
K.Y. Lam and W.M. Ni,
Uniqueness and complete dynamics of the LotkaVolterra competition diffusion system,
SIAM J. Appl. Math., 72 (2012), no.6, 16951712.
(DOI 10.1137/120869481)Read abstractIn this paper we study the interactions between diffusion and heterogeneity of the environment in the classical diffusive LotkaVolterra competition systems. In the weak competition case, we establish the uniqueness, hence the global asymptotic stability, of coexistence steady states under various circumstances, and thereby we obtain a complete understanding of the change in dynamics when one of the interspecific competition coefficients is smal 

3 
X. Chen, K.Y. Lam and Y. Lou,
Dynamics of a reactiondiffusionadvection model for two competing species,
Discrete Contin. Dyn. Syst. A, 32 (2012), 38413859.
(DOI 10.3934/dcds.2012.32.3841) Read abstractWe study the dynamics of a reactiondiffusionadvection model for two competing species in a spatially heterogeneous environment. The two species are assumed to have the same population dynamics but different dispersal strategies: both species disperse by random diffusion and advection along the environmental gradient, but with different random dispersal and/or advection rates. Given any advection rates, we show that three scenarios can occur: (i) If one random dispersal rate is small and the other is large, two competing species coexist; (ii) If both random dispersal rates are large, the species with much larger random dispersal rate is driven to extinction; (iii) If both random dispersal rates are small, the species with much smaller random dispersal rate goes to extinction. Our results suggest that if both advection rates are positive and equal, an intermediate random dispersal rate may evolve. This is in contrast to the case when both advection rates are zero, where the species with larger random dispersal rate is always driven to extinction. 
Corrig. 
2 
K.Y. Lam and W.M. Ni,
Limiting profiles of semilinear elliptic equations with large advection in population dynamics, Special Volume in Honor of Louis Nirenberg's 85th Birthday,
Discrete Contin. Dyn. Syst. A 28 (2010), no. 3, 10511067.
(DOI 10.3934/dcds.2010.28.1051) Read abstractLimiting profiles of solutions to a 2X2 LotkaVolterra competitiondiffusionadvection system, when the strength of the advection tends to infinity, are determined. The two species, competing in a heterogeneous environment, are identical except for their dispersal strategies: One is just random diffusion while the other is smarter  a combination of random diffusion and a directed movement up the environmental gradient. With important progress made, it has been conjectured in [2] and [3] that for large advection the smarter species will concentrate near a selected subset of positive local maximum points of the environment function. In this paper, we establish this conjecture in one space dimension, with the peaks located and the limiting profiles determined, under mild hypotheses on the environment function. 

1 
K.Y. Lam,
Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model,
J. Differential Equations, 250 (2011), no. 1, 161181.
(DOI 10.1016/j.jde.2010.08.028) Read abstractWe consider a reactiondiffusionadvection equation arising from a biological model of migrating species. The qualitative properties of the globally attracting solution are studied and in some cases the limiting profile is determined. In particular, a conjecture of Cantrell, Cosner and Lou on concentration phenomena is resolved under mild conditions. Applications to a related parabolic competition system are also discussed. 
Work in Progress:
1. (With Xinfu Chen, Qing Li and Yaping Wu) Propagation Phenomena for a Nonlocal Reactiondiffusion Model with Bounded Phenotypic Traits.
2. (With ChangHong Wu) Propagation in shifting climate.
3. (With Xiao Yu) The Fluxlimited solution approach to propagation of in KPP equation.
4. (With Lin Wang and Frithjof Lutscher) Effect of diffusion and advection on the principal eigenvalue of cooperative systems in one dimension.
5. (With Chris Henderson) The HamiltonJacobi approach to the propagation in the road and field system.
6. (With C. Cosner and RS. Cantrell) Mean field game and the ideal free distribution.
7. (submitted)(with X.Q. Zhao and M. Zhu) Global dynamics of reactiondiffusion systems with a timevarying domain.
Notes:
1. On Cooperative elliptic systems: Principal eigenvalue and characterization of the maximum principle. (PDF)
Read abstract
This is a note based on my lecture at Center for PDE in East China Normal Unviersity. In this note, the theory of principal eigenvalue of cooperative systems via the KreinRutman Theorem, as well as the characterization of maximum principle for such systems are developed.
2. A note on existence of complete Lyapunov function. (PDF)
Read abstract
In this note we presents a selfcontained proof for the existence of complete Lyapunov function for semiflow admitting a Morse decomposition. The main references are C. Conley's CBMS lecture notes and the monograph by K.P. Rybakowski.
3. A note on critical slope lemmas due to Imbert & Monneau. (PDF)
Read abstract
In this note we presents a proof for the critical slope lemmas which is highly useful in the theory of fluxlimited solution for HamiltonJacobi equations. The proof is taken directly from [ImbertMonneau, 2017].