# American Institute of Mathematical Sciences

## Journals

DCDS
Discrete & Continuous Dynamical Systems - A 2009, 25(1): 123-132 doi: 10.3934/dcds.2009.25.123
We study the long-time behavior of positive solutions to the problem

$u_t-\Delta u=a u-b(x)u^p \mbox{ in } (0,\infty)\times \Omega, Bu=0 \mbox{ on } (0,\infty)\times \partial \Omega,$

where $a$ is a real parameter, $b\geq 0$ is in $C^\mu(\bar{\Omega})$ and $p>1$ is a constant, $\Omega$ is a $C^{2+\mu}$ bounded domain in $R^N$ ($N\geq 2$), the boundary operator $B$ is of the standard Dirichlet, Neumann or Robyn type. Under the assumption that $\overline\Omega_0$:=$\{x\in\Omega: b(x)=0\}$ has non-empty interior, is connected, has smooth boundary and is contained in $\Omega$, it is shown in [8] that when $a\geq \lambda_1^D(\Omega_0)$, for any fixed $x\in \overline{\Omega}_0$, $\overline{\lim}_{t\to\infty}u(t,x)$=$\infty$, and for any fixed $x\in \overline{\Omega}\setminus \overline{\Omega}_0$,

$\overline{\lim}_{t\to\infty}u(t,x)\leq \overline{U}_a(x), \underline{\lim}_{t\to\infty}u(t,x)\geq \underline{U}_a(x), where$\underline{U}_a$and$\overline{U}_a$denote respectively the minimal and maximal positive solutions of the boundary blow-up problem$-\Delta u=au-b(x)u^p \mbox{ in} \ \Omega\setminus\overline{\Omega}_0,\ Bu=0 \mbox{ on}\ \partial \Omega,\ \ u=\infty \mbox{ on}\ \partial \Omega_0.$The main purpose of this paper is to show that, under the above assumptions,$\lim_{t\to\infty} u(t,x)=\underline U_a(x),\forall x\in \overline\Omega\setminus \overline\Omega_0.$This proves a conjecture stated in [8]. Some extensions of this result are also discussed. keywords: DCDS-B Discrete & Continuous Dynamical Systems - B 2017, 22(3): 895-911 doi: 10.3934/dcdsb.2017045 We give a complete description of the long-time asymptotic profile of the solution to a free boundary model considered recently in [10]. This model describes the spreading of an invasive species in an environment which shifts with a constant speed, and the research of [10] indicates that the species may vanish, or spread successfully, or fall in a borderline case.In the case of successful spreading, the long-time behavior of the population is not completely understood in [10].Here we show that the spreading of the species is governed by two traveling waves, one has the speed of the shifting environment, giving the profile of the retreating tail of the population, while the other has a faster speed determined by a semi-wave, representing the profile of the advancing front of the population. keywords: DCDS Discrete & Continuous Dynamical Systems - A 2006, 14(1): 1-29 doi: 10.3934/dcds.2006.14.1 We study the degenerate logistic model described by the equation$ u_t - $Δ$ u=au-b(x)u^p$with standard boundary conditions, where$p>1$,$b$vanishes on a nontrivial subset$\Omega_0$of the underlying bounded domain$\Omega\subset R^N$and$b$is positive on$\Omega_+=\Omega\setminus \overline{\Omega}_0$. We consider the difficult case where$\partial\Omega_0\cap \partial \Omega$≠$\emptyset$and$\partial\Omega_+\cap \partial \Omega$≠$\emptyset$, and examine the asymptotic behaviour of the solutions. By a detailed study of a singularly mixed boundary blow-up problem, we obtain some basic results on the dynamics of the model. keywords: DCDS-B Discrete & Continuous Dynamical Systems - B 2014, 19(10): 3105-3132 doi: 10.3934/dcdsb.2014.19.3105 In this paper we consider the diffusive competition model consisting of an invasive species with density$u$and a native species with density$v$, in a radially symmetric setting with free boundary. We assume that$v$undergoes diffusion and growth in$\mathbb{R}^N$, and$u$exists initially in a ball${r < h(0)}$, but invades into the environment with spreading front${r = h(t)}$, with$h(t)$evolving according to the free boundary condition$h'(t) = -\mu u_r(t, h(t))$, where$\mu>0$is a given constant and$u(t,h(t))=0$. Thus the population range of$u$is the expanding ball${r < h(t)}$, while that for$v$is$\mathbb{R}^N$. In the case that$u$is a superior competitor (determined by the reaction terms), we show that a spreading-vanishing dichotomy holds, namely, as$t\to\infty$, either$h(t)\to\infty$and$(u,v)\to (u^*,0)$, or$\lim_{t\to\infty} h(t)<\infty$and$(u,v)\to (0,v^*)$, where$(u^*,0)$and$(0, v^*)$are the semitrivial steady-states of the system. Moreover, when spreading of$u$happens, some rough estimates of the spreading speed are also given. When$u$is an inferior competitor, we show that$(u,v)\to (0,v^*)$as$t\to\infty$, so the invasive species$u$always vanishes in the long run. keywords: DCDS-S Discrete & Continuous Dynamical Systems - S 2018, 0(0): 1807-1833 doi: 10.3934/dcdss.2019119 In this article, a sample of Norman Dancer's published works are briefly described, to give the reader a taste of his deep and important research on nonlinear functional analysis, nonlinear ODE and PDE problems, and dynamical systems. The sample covers a variety of topics where Norman Dancer has made remarkable contributions. The author takes this opportunity to express his deep admiration of the work of Professor Norman Dancer, and to thank him for the kind help to the development of the author's career, which has been greatly influenced by him and his work. keywords: DCDS Discrete & Continuous Dynamical Systems - A 2007, 19(2): 271-298 doi: 10.3934/dcds.2007.19.271 We show that for small$\epsilon>0$, the boundary blow-up problem$-\epsilon^2\Delta u= u (u-a(x))(1-u) \mbox{ in } \Omega, u|_{\partial\Omega}=\infty\$

has solutions with sharp interior layers and spikes, apart from boundary layers. We also determine the location of these layers and spikes.

keywords:
NHM
Networks & Heterogeneous Media 2012, 7(4): 583-603 doi: 10.3934/nhm.2012.7.583
We investigate, from a more ecological point of view, a free boundary model considered in [11] and [8] that describes the spreading of a new or invasive species, with the free boundary representing the spreading front. We derive the free boundary condition by considering a "population loss" at the spreading front, and correct some mistakes regarding the range of spreading speed in [11]. Then we use numerical simulation to gain further insights to the model, which may help to determine its usefulness in concrete ecological situations.
keywords:
DCDS
Discrete & Continuous Dynamical Systems - A 2008, 21(1): i-ii doi: 10.3934/dcds.2008.21.1i
Professor Edward Norman Dancer, known to his friends and colleagues as Norm or Norman, was born in Bundaberg in north Queensland, Australia in December 1946. He graduated from the Australian National University in 1968 with first class honours, and continued to obtain a PhD from the University of Cambridge in 1972. He was appointed a Lecturer in 1973 at the University of New England, Armidale, where he received a Personal Chair in 1987. He left Armidale in 1993 to become a Professor of Mathematics at the University of Sydney, a position he has held since. He was elected a Fellow of the Australian Academy of Science (FAA) in 1996. He has held distinguished visiting professorships at many institutions in Europe and North America. In 2002 he received the prestigious Alexander von Humboldt Research Award, the highest prize awarded in Germany to foreign scientists.