Research UNEhttps://rune.une.edu.au/webThe Research UNE digital repository system captures, stores, indexes, preserves, and distributes digital research material.Wed, 09 Oct 2024 09:43:16 GMT2024-10-09T09:43:16Z501381Remarks on the uniqueness problem for the logistic equation on the entire spacehttps://hdl.handle.net/1959.11/4251Title: Remarks on the uniqueness problem for the logistic equation on the entire space
Contributor(s): Du, Yihong; Liu, Lishan
Abstract: We show that this problem has a unique positive solution. This considerably improves some earlier results. The main new technique here is a Safonov type iteration argument. The result can also be proved by a technique introduced by Marcus and Veron, and the two different techniques are compared.
Sun, 01 Jan 2006 00:00:00 GMThttps://hdl.handle.net/1959.11/42512006-01-01T00:00:00ZThe Diffusive Competition Model With A Free Boundary: Invasion Of A Superior Or Inferior Competitorhttps://hdl.handle.net/1959.11/17070Title: The Diffusive Competition Model With A Free Boundary: Invasion Of A Superior Or Inferior Competitor
Contributor(s): Du, Yihong; Lin, Zhigui
Abstract: 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.
Wed, 01 Jan 2014 00:00:00 GMThttps://hdl.handle.net/1959.11/170702014-01-01T00:00:00ZQuasilinear Elliptic Equations on Half- and Quarter-spaceshttps://hdl.handle.net/1959.11/12820Title: Quasilinear Elliptic Equations on Half- and Quarter-spaces
Contributor(s): Dancer, Edward Norman; Du, Yihong; Efendiev, Messoud
Abstract: We consider quasilinear elliptic problems of the form... Our results extend most of those in the recent paper of Efendiev and Hamel [6] for the special case p = 2 to the general case p > 1. Moreover, by making use of a sharper Liouville type theorem, some of the results in [6] are improved. To overcome the difficulty of the lack of a strong comparison principle for p-Laplacian problems, we employ a weak sweeping principle.
Tue, 01 Jan 2013 00:00:00 GMThttps://hdl.handle.net/1959.11/128202013-01-01T00:00:00ZSpreading Profile and Nonlinear Stefan Problemshttps://hdl.handle.net/1959.11/17084Title: Spreading Profile and Nonlinear Stefan Problems
Contributor(s): Du, Yihong
Abstract: We report some recent progress on the study of the following nonlinear Stefan problem ut − ∆u = f(u) for x ∈ Ω(t), t > 0, u = 0 and ut = µ|∇xu| 2 for x ∈ Γ(t), t > 0, u(0, x) = u0(x) for x ∈ Ω0, where Ω(t) ⊂ R N (N ≥ 1) is bounded by the free boundary Γ(t), with Ω(0) = Ω0, µ is a given positive constant. The initial function u0 is positive in Ω0 and vanishes on ∂Ω0. The class of nonlinear functions f(u) includes the standard monostable, bistable and combustion type nonlinearities. When µ → ∞, it can be shown that this free boundary problem converges to the corresponding Cauchy problem ut − ∆u = f(u) for x ∈ R N , t > 0, u(0, x) = u0(x) for x ∈ R N . We will discuss the similarity and differences of the dynamical behavior of these two problems by closely examining their spreading profiles, which suggest that the Stefan condition is a stabilizing factor in the spreading process.
Tue, 01 Jan 2013 00:00:00 GMThttps://hdl.handle.net/1959.11/170842013-01-01T00:00:00ZSemi-wave and spreading speed of the nonlocal Fisher-KPP equation with free boundarieshttps://hdl.handle.net/1959.11/31893Title: Semi-wave and spreading speed of the nonlocal Fisher-KPP equation with free boundaries
Contributor(s): Du, Yihong; Li, Fang; Zhou, Maolin
Abstract: <p>In Cao, Du, Li and Li [9], a nonlocal diffusion model with free boundaries extending the local diffusion model of Du and Lin [18] was introduced and studied. For Fisher-KPP type nonlinearities, its long-time dynamical behaviour is shown to follow a spreading-vanishing dichotomy. However, when spreading happens, the question of spreading speed was left open in [9]. In this paper we obtain a rather complete answer to this question. We find a threshold condition on the kernel function such that spreading grows linearly in time exactly when this condition holds, which is achieved by completely solving the associated semi-wave problem that determines this linear speed; when the kernel function violates this condition, we show that accelerated spreading happens.</p>
Fri, 01 Oct 2021 00:00:00 GMThttps://hdl.handle.net/1959.11/318932021-10-01T00:00:00ZNonlinear Critical Elliptic Problems: Existence and Uniquenesshttps://hdl.handle.net/1959.11/56258Title: Nonlinear Critical Elliptic Problems: Existence and Uniqueness
Contributor(s): Li, Benniao; Du, Yihong
Abstract: <p>In this thesis, I shall study the existence, local uniqueness and other related subjects for the bubbling solutions of two elliptic problems involving critical Sobolev exponent.</p>
Description: Chancellor's Doctoral Research Medal awarded on 7th June, 2019.
Please contact rune@une.edu.au if you require access to this thesis for the purpose of research or study.
Fri, 07 Jun 2019 00:00:00 GMThttps://hdl.handle.net/1959.11/562582019-06-07T00:00:00ZTwo species nonlocal diffusion systems with free boundarieshttps://hdl.handle.net/1959.11/31904Title: Two species nonlocal diffusion systems with free boundaries
Contributor(s): Du, Yihong; Wang, Mingxin; Zhao, Meng
Abstract: <p>We study a class of free boundary systems with nonlocal diffusion, which are natural extensions of the corresponding free boundary problems of reaction diffusion systems. As before the free boundary represents the spreading front of the species, but here the population dispersal is described by "nonlocal diffusion" instead of "local diffusion". We prove that such a nonlocal diffusion problem with free boundary has a unique global solution, and for models with Lotka-Volterra type competition or predator-prey growth terms, we show that a spreading-vanishing dichotomy holds, and obtain criteria for spreading and vanishing; moreover, for the weak competition case and for the weak predation case, we can determine the long-time asymptotic limit of the solution when spreading happens. Compared with the single species free boundary model with nonlocal diffusion considered recently in [7], and the two species cases with local diffusion extensively studied in the literature, the situation considered in this paper involves several new difficulties, which are overcome by the use of some new techniques.</p>
Tue, 01 Mar 2022 00:00:00 GMThttps://hdl.handle.net/1959.11/319042022-03-01T00:00:00ZSpreading speed for some cooperative systems with nonlocal diffusion and free boundaries, part 1: Semi-wave and a threshold conditionhttps://hdl.handle.net/1959.11/32078Title: Spreading speed for some cooperative systems with nonlocal diffusion and free boundaries, part 1: Semi-wave and a threshold condition
Contributor(s): Du, Yihong; Ni, Wenjie
Abstract: <p>We consider a class of cooperative reaction-diffusion systems with free boundaries in one space dimension, where the diffusion terms are nonlocal, given by integral operators involving suitable kernel functions, and they are allowed not to appear in some of the equations in the system. Such a system covers various models arising from mathematical biology, in particular a West Nile virus model and an epidemic model considered recently in [16] and [44], respectively, where a "spreading-vanishing" dichotomy is known to govern the long time dynamical behaviour, but the question on spreading speed was left open. In this paper, we develop a systematic approach to determine the spreading profile of the system, and obtain threshold conditions on the kernel functions which decide exactly when the spreading has finite speed, or infinite speed (accelerated spreading). This relies on a rather complete understanding of both the associated semi-waves and travelling waves. When the spreading speed is finite, we show that the speed is determined by a particular semi-wave. This is Part 1 of a two part series. In Part 2, for some typical classes of kernel functions, we will obtain sharp estimates of the spreading rate for both the finite speed case, and the infinite speed case.</p>
Tue, 25 Jan 2022 00:00:00 GMThttps://hdl.handle.net/1959.11/320782022-01-25T00:00:00ZOn elements of the analysis and modelling of collective motionhttps://hdl.handle.net/1959.11/53544Title: On elements of the analysis and modelling of collective motion
Contributor(s): Mudaliar, Rajnesh Krishnan; Schaerf, Timothy; Du, Yihong
Abstract: <p>Groups of animals coordinate remarkable, coherent, movement patterns during periods of collective motion. Such movement patterns include the toroidal mills seen in fish shoals, highly aligned parallel motion like that of flocks of migrating birds, and the swarming of insects. Since the 1970’s a wide range of collective motion models have been studied that prescribe rules of interaction between individuals, and that are capable of generating emergent patterns that are visually similar to those seen in real animal groups. This does not necessarily mean that real animals apply exactly the same interactions as those prescribed in models. In more recent work, researchers have sought to infer the rules of interaction of real animals directly from tracking data, by using a number of techniques, including force mapping methods. In one of the simplest formulations, the force mapping methods determine the mean changes in the components of the velocity of an individual over time as a function of the relative coordinates of group mates. The force mapping methods can also be modified to estimate other closely related quantities including the mean relative direction of motion of group mates as a function of their relative coordinates. Since these methods for extracting interaction rules and related quantities from trajectory data are relatively new, the accuracy of these methods has had limited inspection. In my thesis, I examine the ability of the force mapping method to reveal prescribed rules of interaction from data generated by three individual based models for collective motion, namely the zonal model developed by Couzin et al. [20], the ODE model developed by D’Orsogna et al. [22] and an alignment only model developed by Vicsek et al. [87], as well as variants of the Couzin et al. [20] and Vicsek et al. [87] models where some interactions apply over topological scales. Topological scales refer to the distance-based neighbour rank of other members of a group, in contrast to metric scales, which are based on linear distances.</p> <p>My analysis suggests that force maps constructed in a standard form (where mean changes in components of individuals’ velocity are mapped as a function of the relative positions of neighbours) capture the qualitative, and sometimes quantitative, features of interaction rules including repulsion and attraction effects, and the presence of blind zones. However, the features of standard force maps may also be affected by emergent group level patterns of movement, and the sizes of the regions over which repulsion and attraction effects are apparent can be distorted as group size varies, dependent on how individuals respond when interacting with multiple neighbours.</p> <p>I also examined the effectiveness of force maps tailored to examine orientation/alignment interactions, and interactions that apply over topological scales, via appropriate choices of dependent variables including: differences in directions of motion between a focal individual and its neighbour (angular differences), relative positions of, or distances to, neighbours, and the distance-based rank of neighbours. Force maps that illustrated mean changes in direction as a function of metric and/or topological distances to, and angular differences with, neighbours can provide relatively clear and consistent information about the presence of a direct underlying tendency for individuals to align with neighbours. However, these force maps also tend to overestimate the domain over which orientation/alignment interactions apply, as compared to explicit interactions between pairs. Force maps of this type that take into account the relative position of neighbours, rather than just their distance from an individual, can also reveal orientation behaviour, but often less clearly. In addition, such more detailed force maps seem to be strongly affected by emergent patterns of motion.</p> <p>In related analysis, I examined the relationship between regions of high alignment in local alignment plots (plots that illustrate statistics of the relative directions of neighbours as a function of their relative coordinates), and the prescribed regions over which orientation interaction rules applied. In the case that simulated individuals only applied orientationbased rules for adjusting their velocity, regions of high alignment were correlated in size with prescribed orientation zones. However, this relationship did not hold when data from a more complex model was analysed, and a region of high alignment could appear in a local alignment plot even when there was no underlying behavioural rule for orientation. Ultimately, this suggests that local alignment plots are reflective of emergent behaviour, but not necessarily of the underlying interactions driving such behaviour.</p> <p>Following on from my investigation of force maps, I then applied some of these maps to examine potential orientation interactions in small shoals of fish of three different species (Xray tetras, eastern mosquitofish and crimson spotted rainbowfish) from existing experimental data sets. The analysis suggested that all three species have an explicit underlying behavioural rule for aligning their movements with neighbours.</p> <p>The models used throughout my thesis exhibit multiple forms of parameter and initial condition dependent emergent patterns of collective motion. In the final portion of my thesis, I examine the emergent states of one of these models - the aforementioned modification of the Couzin et al. [20] model in three spatial dimensions. The model was modified such that interactions relating to aligning directions of motion (orientation interactions) and moving towards more distant group mates (attraction interactions) operated over topological (distance based neighbour rank) scales, rather than metric (linear distance) scales. Collision avoidance (repulsion) interactions operated over a metric scale, close to each individual, as in the original model. I examined the emergent group level patterns of movement generated by the model as the numbers of neighbours that contributed to orientation and attraction based adjustments to velocity were varied. Like the metric form of the model, simulated groups could fragment, move in a swarm-like manner, move together in parallel, and mill. However, milling was extremely rare, with emergent states classified as milling not necessarily exhibiting the same clearly ordered rotational structure as typical examples from the metric form of the model. The model also generated other cohesive group movements that were not easily classified in terms of swarming, milling, or aligned motion, and in some cases these groups exhibited directed motion without strong alignment of individuals. Groups that did not fragment tended to stay relatively compact in terms of both mean and nearest neighbour distances. Even if a group did fragment, individuals remained relatively close to near neighbours, avoiding complete isolation.</p> <p>This broad study attempts to provide greater insight into the accuracy of force maps and some of the factors that may contribute to their inaccuracy, and highlight some of the richness behind fundamental methods for analysing and modelling collective motion.</p>
Wed, 24 Nov 2021 00:00:00 GMThttps://hdl.handle.net/1959.11/535442021-11-24T00:00:00ZSpreading under shifting climate by a free boundary model: Invasion of deteriorated environmenthttps://hdl.handle.net/1959.11/38744Title: Spreading under shifting climate by a free boundary model: Invasion of deteriorated environment
Contributor(s): Hu, Yuanyang; Hao, Xinan; Du, Yihong
Abstract: <p>In this paper, we consider a free boundary model in one space dimension which describes the spreading of a species subject to climate change, where favorable environment is shifting away with a constant speed c > 0 and replaced by a deteriorated yet still favorable environment. We obtain two threshold speeds c<sub>1</sub> < c<sub>0</sub> and a complete classification of the long-time dynamics of the model, which reveals significant differences between the cases 0 < c < c<sub>1</sub>, c = c<sub>1</sub>, c<sub>1</sub> < c < c<sub>0</sub> and c ≥ c<sub>0</sub>. For example, when c<sub>1</sub> < c < c<sub>0</sub>, for a suitably parameterized family of initial functions <i>u</i>σ0 increasing continuously in σ, we show that there exists 0 < σ<sub>⁎</sub> < σ<sup>⁎</sup> < ∞ such that the species vanishes eventually when σ ∈ (0, σ<sub>⁎</sub>], it spreads with asymptotic speed c<sub>1</sub> when σ ∈ (σ<sub>⁎</sub>, σ<sup>⁎</sup>), it spreads with forced speed c when σ = σ<sup>⁎</sup>, and it spreads with speed c<sub>0</sub> when σ > σ<sup>⁎</sup>. Moreover, in the last case, while the spreading front propagates with asymptotic speed c<sub>0</sub>, the profile of the population density function <i>u</i>(<i>t, x</i>) approaches a propagating pair consisting of a traveling wave with speed c and a semi-wave with speed c<sub>0</sub>.</p>
Wed, 01 Dec 2021 00:00:00 GMThttps://hdl.handle.net/1959.11/387442021-12-01T00:00:00ZA diffusive logistic model with a free boundary in time-periodic environmenthttps://hdl.handle.net/1959.11/13629Title: A diffusive logistic model with a free boundary in time-periodic environment
Contributor(s): Du, Yihong; Guo, Zongming; Peng, Rui
Abstract: We study the diffusive logistic equation with a free boundary in time-periodic environment. Such a model may be used to describe the spreading of a new or invasive species, with the free boundary representing the expanding front. For time independent environment, in the cases of one space dimension, and higher space dimensions with radial symmetry, this free boundary problem has been studied in Du and Lin (2010), Du and Guo (2011). In both cases, a spreading-vanishing dichotomy was established, and when spreading occurs, the asymptotic spreading speed was determined. In this paper, we show that the spreading-vanishing dichotomy is retained in time-periodic environment, and we also determine the spreading speed. The former is achieved by further developing the earlier techniques, and the latter is proved by introducing new ideas and methods.
Tue, 01 Jan 2013 00:00:00 GMThttps://hdl.handle.net/1959.11/136292013-01-01T00:00:00ZTraveling waves for a generalized Holling-Tanner predator-prey modelhttps://hdl.handle.net/1959.11/22829Title: Traveling waves for a generalized Holling-Tanner predator-prey model
Contributor(s): Ai, Shangbing; Du, Yihong; Peng, Rui
Abstract: We study traveling wave solutions for Holling-Tanner type predator-prey models, where the predator equation has a singularity at zero prey population. The traveling wave solutions here connect the prey only equilibrium (1, 0)with the unique constant coexistence equilibrium (u∗, v∗). First, we give a sharp existence result on weak traveling wave solutions for a rather general class of predator-prey systems, with minimal speed explicitly determined. Such a weak traveling wave (u(ξ), v(ξ))connects (1, 0)at ξ=-∞but needs not connect (u∗, v∗)at ξ=∞. Next we modify the Holling-Tanner model to remove its singularity and apply the general result to obtain a weak traveling wave solution for the modified model, and show that the prey component in this weak traveling wave solution has a positive lower bound, and thus is a weak traveling wave solution of the original model. These results for weak traveling wave solutions hold under rather general conditions. Then we use two methods, a squeeze method and a Lyapunov function method, to prove that, under additional conditions, the weak traveling wave solutions are actually traveling wave solutions, namely they converge to the coexistence equilibrium as ξ→∞.
Sun, 01 Jan 2017 00:00:00 GMThttps://hdl.handle.net/1959.11/228292017-01-01T00:00:00ZLogistic type equations on RN by a squeezing method involving boundary blow-up solutionshttps://hdl.handle.net/1959.11/213Title: Logistic type equations on RN by a squeezing method involving boundary blow-up solutions
Contributor(s): Du, Y; Ma, L
Abstract: We study, on the entire space RN(N≥1), the diffusive logistic equationut–∆u=λu–up, u≥0 (1.1)and its generalizations. Here p > 1 is a constant. Problem (1.1) plays an important role in understanding various population models and some other problems in applied mathematics. When λ = 1 and p = 2, it is also known as the Fisher equation and KPP equation, due to the pioneering works of Fisher [8] and Kolmogoroff, Petrovsky and Piscounoff [18].
Mon, 01 Jan 2001 00:00:00 GMThttps://hdl.handle.net/1959.11/2132001-01-01T00:00:00ZSharp asymptotic profile of the solution to a West Nile virus model with free boundaryhttps://hdl.handle.net/1959.11/57758Title: Sharp asymptotic profile of the solution to a West Nile virus model with free boundary
Contributor(s): Wang, Zhiguo; Nie, Hua; Du, Yihong
Abstract: <p>We consider the long-time behaviour of a West Nile virus (WNv) model consisting of a reaction–diffusion system with free boundaries. Such a model describes the spreading of WNv with the free boundary representing the expanding front of the infected region, which is a time-dependent interval [<i>g(t), h(t)</i>] in the model (Lin and Zhu, Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary. <i>J. Math. Biol</i>. 75, 1381–1409, 2017). The asymptotic spreading speed of the front has been determined in Wang et al. (Spreading speed for a West Nile virus model with free boundary. <i>J. Math. Biol</i>. 79, 433–466, 2019) by making use of the associated semi-wave solution, namely <i>lim<sub>t→∞</sub> h(t)/t = lim<sub>t→∞</sub> [− g(t)/t] = c<sub>ν</sub></i> , with c<sub>ν</sub></i> the speed of the semi-wave solution. In this paper, by employing new techniques, we significantly improve the estimate in Wang et al. (Spreading speed for a West Nile virus model with free boundary. <i>J. Math. Biol</i>. 79, 433–466, 2019): we show that <i>h(t) − c<sub>ν</sub>t</i> and <i>g(t) + c<sub>ν</sub>t</i> converge to some constants as <i>t → ∞</i>, and the solution of the model converges to the semi-wave solution. The results also apply to a wide class of analogous Ross–MacDonold epidemic models. </p>
Mon, 01 Jan 2024 00:00:00 GMThttps://hdl.handle.net/1959.11/577582024-01-01T00:00:00ZRate of accelerated expansion of the epidemic region in a nonlocal epidemic model with free boundarieshttps://hdl.handle.net/1959.11/57756Title: Rate of accelerated expansion of the epidemic region in a nonlocal epidemic model with free boundaries
Contributor(s): Du, Yihong; Ni, Wenjie; Wang, Rong
Abstract: <p>This paper is concerned with the long-time dynamics of an epidemic model whose diffusion and reaction terms involve nonlocal effects described by suitable convolution operators, and the epidemic region is represented by an evolving interval enclosed by the free boundaries in the model. In Wang and Du (2022<i>J. Differ. Eqn</i>. <b>327</b> 322–81), it was shown that the model is well-posed, and its long-time dynamical behaviour is governed by a spreading-vanishing dichotomy. The spreading speed was investigated in a subsequent work of Wang and Du (2023 <i>Discrete Contin. Dyn. Syst</i>. <b>43</b> 121–61), where a threshold condition for the diffusion kernels <i>J<sub>1</sub></i> and <i>J<sub>2</sub></i> was obtained, such that the asymptotic spreading speed is finite precisely when this condition is satisfied. In this paper, we examine the case that this threshold condition is not satisfied, which leads to accelerated spreading; for some typical classes of kernel functions, we determine the precise rate of accelerated expansion of the epidemic region by constructing delicate upper and lower solutions.</p>
Mon, 18 Sep 2023 00:00:00 GMThttps://hdl.handle.net/1959.11/577562023-09-18T00:00:00ZExact rate of accelerated propagation in the Fisher-KPP equation with nonlocal diffusion and free boundarieshttps://hdl.handle.net/1959.11/57757Title: Exact rate of accelerated propagation in the Fisher-KPP equation with nonlocal diffusion and free boundaries
Contributor(s): Du, Yihong; Ni, Wenjie
Abstract: <p>Accelerated propagation is a new phenomenon associated with nonlocal diffusion problems. In this paper, we determine the exact rate of accelerated propagation in the Fisher-KPP equation with nonlocal diffusion and free boundaries.</p>
https://hdl.handle.net/1959.11/57757Sharp conditions for the existence of boundary blow-up solutions to the Monge–Ampère equationhttps://hdl.handle.net/1959.11/23044Title: Sharp conditions for the existence of boundary blow-up solutions to the Monge–Ampère equation
Contributor(s): Zhang, Xuemei; Du, Yihong
Abstract: In this paper we give sharp conditions on K(x) and f (u) for the existence of strictly convex solutions to the boundary blow-up Monge-Ampère problem M[u](x) = K(x) f (u) for x ∈ Ω, u(x)→+∞ as dist(x, ∂Ω) → 0. Here M[u] = det (uxi x j ) is the Monge-Ampère operator, and Ω is a smooth, bounded, strictly convex domain in RN (N ≥ 2). Further results are obtained for the special case that Ω is a ball. Our approach is largely based on the construction of suitable sub- and super-solutions.
Mon, 01 Jan 2018 00:00:00 GMThttps://hdl.handle.net/1959.11/230442018-01-01T00:00:00ZA Liouville theorem for conformal Gaussian curvature type equations in R²https://hdl.handle.net/1959.11/10543Title: A Liouville theorem for conformal Gaussian curvature type equations in R²
Contributor(s): Du, Yihong; Ma, Li
Abstract: In this paper, we obtain a Liouville type theorem for a class of elliptic equations including the conformal Gaussian curvature equation... We notice that all these previous studies require that K(x) has a fixed sign near infinity or K(x) decays to 0 at infinity. In this paper we are mainly concerned with the case that K(x) does not have these properties.
Sun, 01 Jan 2012 00:00:00 GMThttps://hdl.handle.net/1959.11/105432012-01-01T00:00:00ZPositive solutions of an elliptic equation with negative exponent: stability and critical powerhttps://hdl.handle.net/1959.11/6228Title: Positive solutions of an elliptic equation with negative exponent: stability and critical power
Contributor(s): Du, Yihong; Guo, Zongming
Abstract: We study positive solutions of the equation Δu = |x|α u−p in Ω ⊂ RN (N ≥ 2), where p > 0, α > −2, and Ω is a bounded or unbounded domain. We show that there is a critical power p = pc(α) such that this equation with Ω = ℝN has no stable positive solution for p > pc(α) but it admits a family of stable positive solutions when 0 < p ≤ pc(α). If p > pc(α⁻) (α⁻ = min{α, 0}), we further show that this equation with Ω = Br {0} has no positive solution with finite Morse index that has an isolated rupture at 0, and analogously it has no positive solution with finite Morse index when Ω = ℝN BR . Among other results, we also classify the positive solutions over Br {0} which are not bounded near 0.
Thu, 01 Jan 2009 00:00:00 GMThttps://hdl.handle.net/1959.11/62282009-01-01T00:00:00ZSome Recent Results on Diffusive Predator-prey Models in Spatially Heterogeneous Environmenthttps://hdl.handle.net/1959.11/2629Title: Some Recent Results on Diffusive Predator-prey Models in Spatially Heterogeneous Environment
Contributor(s): Du, Yihong; Shi, Junping
Abstract: We present several recent results obtained in our attempts to understand the influence of spatial heterogeneity in the predator-prey models. Two different approaches are taken. The first approach is based on the observation that the behavior of many diffusive population models is very sensitive to certain coefficient functions becoming small in part of the underlying spatial region. We apply this observation to three predator-prey models to reveal fundamental differences from the classical homogeneous case in each model, and demonstrate the essential differences of these models from each other. In the second approach, we examine the influence of a protection zone in a Holling type II diffusive predator-pre model, which introduces different mathematical problems from those in the first approach, and reveals important impacts of the protection zone.
Sun, 01 Jan 2006 00:00:00 GMThttps://hdl.handle.net/1959.11/26292006-01-01T00:00:00ZUniqueness and Layer Analysis for Boundary Blow-up Solutionshttps://hdl.handle.net/1959.11/434Title: Uniqueness and Layer Analysis for Boundary Blow-up Solutions
Contributor(s): Du, Y; Guo, Z
Abstract: We prove uniqueness for boundary blow-up solutions of the problem:Δu=λf(u) in Ω,u|∂Ω=∞, with large λ. Previous uniqueness results require a monotonicity assumption for f(u)lu in the entire range of the boundary blow-up solutions. By obtaining good boundary layer estimates for large λ, we obtain uniqueness under much weaker assumptions on f(u). Our estimates for the layers of the boundary blow-up solutions have independent interest, and may have other applications.
Thu, 01 Jan 2004 00:00:00 GMThttps://hdl.handle.net/1959.11/4342004-01-01T00:00:00ZSign Changing Solutions with Clustered Layers Near the Origin for Singularly Perturbed Semilinear Elliptic Problems On a Ballhttps://hdl.handle.net/1959.11/5465Title: Sign Changing Solutions with Clustered Layers Near the Origin for Singularly Perturbed Semilinear Elliptic Problems On a Ball
Contributor(s): Du, Yihong; Liu, Zhaoli; Pistoia, Angela; Yan, Shusen
Abstract: We study sign changing solutions to equations of the form -∊²∆u+u=f(u) in B, ∂v u=0 on ∂B, where B is the unit ball in ℝ^N (N≥2), ∊ is a positive constant and f(u) behaves like |u|^p-1 u (but not necessarily odd) with 1 < p < (N+2)/(N-2) if N≥3, and 1 < p <∞ if N=2. We show that for any given positive integer n, this problem has a sign changing radial solution v∊(|x|) which changes sign at exactly n spheres ⋃^n/j=1 {|x|=ρ∊/j}, where 0 < ρ∊/1 < ⋯ < ρ∊/n < 1 and as ∊→0, ρ∊/j→0 and v∊→0 uniformly on compact subsets of (0,1]. Moreover, given any sequence ∊k→0, there is a subsequence ∊ki, such that u∊(|x|) converges to some U in C¹loc(ℝ^N) along this subsequence, and U=U(|x|) is a radial sign changing solution of -∆U+U=f(U) in ℝ^N, U ∈ H¹(ℝ^N) with exactly n zeros: 0 < ρ₁ < ⋯ < ρn < ∞, and ∊⁻¹ρ∊/j→pj along the subsequence ∊ki. Hence the sharp layers of the sign changing solution v∊ are clustered near the origin. The same result holds if the Neumann boundary condition is replaced by the Dirichlet boundary condition, or if B is replaced by ℝ^N.
Tue, 01 Jan 2008 00:00:00 GMThttps://hdl.handle.net/1959.11/54652008-01-01T00:00:00ZSharp spatiotemporal patterns in the diffusive time-periodic logistic equationhttps://hdl.handle.net/1959.11/13905Title: Sharp spatiotemporal patterns in the diffusive time-periodic logistic equation
Contributor(s): Du, Yihong; Peng, Rui
Abstract: To reveal the complex influence of heterogeneous environment on population systems, we examine the asymptotic profile (as ϵ → 0) of the positive solution to the perturbed periodic logistic equation... We show that the temporal degeneracy of 'b' induces sharp spatiotemporal patterns of the solution only when spatial degeneracy also exists; but in sharp contrast, whether or not temporal degeneracy appears in 'b', the spatial degeneracy always induces sharp spatiotemporal patterns, though they differ significantly according to whether temporal degeneracy is also present.
Tue, 01 Jan 2013 00:00:00 GMThttps://hdl.handle.net/1959.11/139052013-01-01T00:00:00ZFinite Morse-Index Solutions and Asymptotics of Weighted Nonlinear Elliptic Equationshttps://hdl.handle.net/1959.11/13904Title: Finite Morse-Index Solutions and Asymptotics of Weighted Nonlinear Elliptic Equations
Contributor(s): Du, Yihong; Guo, Zongming
Abstract: By introducing a suitable setting, we study the behavior of finite Morse-index solutions of the equation... We show that under our chosen setting for the finite Morse-index theory of (1), the stability of a solution to (1) is unchanged under various natural transformations. This enables us to reveal two critical values of the exponent p in (1) that divide the behavior of finite Morse-index solutions of (1), which in turn yields two critical powers for (2) through the transformation. The latter appear difficult to obtain by working directly with (2).
Tue, 01 Jan 2013 00:00:00 GMThttps://hdl.handle.net/1959.11/139042013-01-01T00:00:00ZSharp Estimate of the Spreading Speed Determined by Nonlinear Free Boundary Problemshttps://hdl.handle.net/1959.11/18244Title: Sharp Estimate of the Spreading Speed Determined by Nonlinear Free Boundary Problems
Contributor(s): Du, Yihong; Matsuzawa, Hiroshi; Zhou, Maolin
Abstract: We study nonlinear diffusion problems of the form ut = uxx + f(u) with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundaries representing the expanding fronts. For monostable, bistable, and combustion types of nonlinearities, Du and Lou ["Spreading and vanishing in nonlinear diffusion problems with free boundaries," J. Eur. Math. Soc. (JEMS), to appear] obtained a rather complete description of the long-time dynamical behavior of the problem and revealed sharp transition phenomena between spreading (limt→∞u(t, x) = 1) and vanishing (limt→∞ u(t, x) = 0). They also determined the asymptotic spreading speed of the fronts by making use of semiwaves when spreading happens. In this paper, we give a much sharper estimate for the spreading speed of the fronts than that in the above-mentioned work of Du and Lou, and we describe how the solution approaches the semiwave when spreading happens.
Wed, 01 Jan 2014 00:00:00 GMThttps://hdl.handle.net/1959.11/182442014-01-01T00:00:00ZEffects of a Degeneracy in the Competition Model Part II.: Perturbation and Dynamical Behaviourhttps://hdl.handle.net/1959.11/453Title: Effects of a Degeneracy in the Competition Model Part II.: Perturbation and Dynamical Behaviour
Contributor(s): Du, Y
Abstract: This is the second part of our study on the competition model where the coefficient functions are strictly positive over the underlying spatial region Ω except b(x), which vanishes in a nontrivial subdomain of Ω, and is positive in the rest of Ω. In part I, we mainly discussed the existence of two kinds of steady-state solutions of this system, namely, the classical steady-states and the generalized steady-states. Here we use these solutions to determine the dynamics of the model. We do this with the help of the perturbed model where b(x) is replaced by b(x)+ε, which itself is a classical competition model. This approach also reveals the interesting relationship between the steady-state solutions (both classical and generalized) of the above system and that of the perturbed system.
Tue, 01 Jan 2002 00:00:00 GMThttps://hdl.handle.net/1959.11/4532002-01-01T00:00:00ZEffects of a Degeneracy in the Competition Model Part I.: Classical and Generalized Steady-State Solutionshttps://hdl.handle.net/1959.11/457Title: Effects of a Degeneracy in the Competition Model Part I.: Classical and Generalized Steady-State Solutions
Contributor(s): Du, Y
Abstract: We study the competition model where the coefficient functions are strictly positive over the underlying spatial region Ω except b(x), which vanishes in a nontrivial subdomain of Ω, and is positive in the rest of Ω. We show that there exists a critical number λ* such that if λ <λ*, then the model behaves similarly to the well-studied classical competition model where all the coefficient functions are positive constants, but when λ>λ*, new phenomena occur. Our results demonstrate the fact that heterogeneous environmental effects on population models are not only quantitative, but can be qualitative as well. In part I here, we mainly study two kinds of steady-state solutions which determine the dynamics of the model: one consists of finite functions while the other consists of generalized functions which satisfy (u, v)=(∞, 0) on the part of the domain that b(x) vanishes, but are positive and finite on the rest of the domain, and are determined by certain boundary blow-up systems. The research is continued in part II, where these two kinds of steady-state solutions will be used to determine the dynamics of the model.
Tue, 01 Jan 2002 00:00:00 GMThttps://hdl.handle.net/1959.11/4572002-01-01T00:00:00ZPositive solutions of elliptic equations with a strong singular potentialhttps://hdl.handle.net/1959.11/28572Title: Positive solutions of elliptic equations with a strong singular potential
Contributor(s): Wei, Lei; Du, Yihong
Abstract: <p>In this paper, we study positive solutions of the elliptic equation
</p><p>
−Δ𝑢=𝜆𝑑(𝑥)𝛼𝑢−𝑑(𝑥)𝜎𝑢𝑝inΩ,
</p><p>
where 𝛼>2,𝜎>−𝛼, 𝑝>1, 𝑑(𝑥)=𝑑𝑖𝑠𝑡(𝑥,𝜕Ω), and Ω is a bounded smooth domain in ℝ𝑁(𝑁⩾2). When 𝛼=2, the term 1𝑑(𝑥)𝛼=1𝑑(𝑥)2 is often called a Hardy potential, and the equation in this case has been extensively investigated. Here we consider the case 𝛼>2, which gives a stronger singularity than the Hardy potential near 𝜕Ω. We show that when 𝜆<0, the equation has no positive solution, while when 𝜆>0, the equation has a unique positive solution, and it satisfies
</p><p>
lim𝑑(𝑥)→0𝑢(𝑥)𝑑(𝑥)𝛼+𝜎𝑝−1=𝜆1𝑝−1.</p>
Tue, 01 Jan 2019 00:00:00 GMThttps://hdl.handle.net/1959.11/285722019-01-01T00:00:00ZSpreading and vanishing in nonlinear diffusion problems with free boundarieshttps://hdl.handle.net/1959.11/18389Title: Spreading and vanishing in nonlinear diffusion problems with free boundaries
Contributor(s): Du, Yihong; Lou, Bendong
Abstract: We study nonlinear diffusion problems of the form ut = uxx + f (u) with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For special f (u) of the Fisher-KPP type, the problem was investigated by Du and Lin [DL]. Here we consider much more general nonlinear terms. For any f (u) which is Ϲ¹ and satisfies f (0) = 0, we show that the omega limit set ω(u) of every bounded positive solution is determined by a stationary solution. For monostable, bistable and combustion types of nonlinearities, we obtain a rather complete description of the long-time dynamical behavior of the problem; moreover, by introducing a parameter σ in the initial data, we reveal a threshold value σ* such that spreading (limt→∞u = 1) happens when σ > σ*, vanishing (limt→∞u = 0) happens when σ < σ*, and at the threshold value σ*, ω(u) is different for the three different types of nonlinearities. When spreading happens, we make use of "semi-waves" to determine the asymptotic spreading speed of the front.
Thu, 01 Jan 2015 00:00:00 GMThttps://hdl.handle.net/1959.11/183892015-01-01T00:00:00ZNonlinear Diffusion Problems With Free Boundaries: Convergence, Transition Speed, and Zero Number Argumentshttps://hdl.handle.net/1959.11/18390Title: Nonlinear Diffusion Problems With Free Boundaries: Convergence, Transition Speed, and Zero Number Arguments
Contributor(s): Du, Yihong; Lou, Bendong; Zhou, Maolin
Abstract: This paper continues the investigation of [Du and Lou, 'J. Eur. Math. Soc. (JEMS)', arXiv:1301.5373, 2013], where the long-time behavior of positive solutions to a nonlinear diffusion equation of the form 'ut = uₓₓ + f(u)' for 'x' over a varying interval '(g(t), h(t))' was examined.
Thu, 01 Jan 2015 00:00:00 GMThttps://hdl.handle.net/1959.11/183902015-01-01T00:00:00ZLocally Uniform Convergence to an Equilibrium for Nonlinear Parabolic Equations on ℝNhttps://hdl.handle.net/1959.11/18391Title: Locally Uniform Convergence to an Equilibrium for Nonlinear Parabolic Equations on ℝN
Contributor(s): Du, Yihong; Polacik, Peter
Abstract: <p>We consider bounded solutions of the Cauchy problem</p> <table><tr><td>{</td><td><i>U<sub>t</sub></i> - Δ<i>u</i> = ƒ<sub><i>u</sub>,</td><td> x</i> ∈ℝ<i><sup>N</sup>, t</i> > 0,</td></tr><tr><td> </td><td><i>u</i>(O, <i>x</i>) = <i>u</i><sub>o</sub> (<i>x</i>), </td><td> x</i> ∈ℝ<i><sup>N</sup>,</td></tr></table> </p>where <i>u</i><sub>0</sub> is a non-negative function with compact support and ƒ is a <i>C</i><sup>1</sup> function on ℝ with ƒ (O) = 0. Assuming that ƒ' is locally Hölder continuous, and that ƒ satisfies minor nondegeneracy condition we prove that as <i>t</i> → ∞ the solution <i>u(∙, t</i>) converges to an equilibriun <i>φ</i> locally uniformly in ℝ<i><sup>N</sup></i>. Moreover, either the limit function <i>φ</i> is a constant equilibrium, or there is a point <i>x</i><sub>o</sub> ∈ ℝ<i><sup>N</sup></i> such that <i>φ</i> is radically symmetric and radially decreasing about <i>x</i><sub>o</sub>, and it approaches a constant equilibrium as |<i>x - x</i><sub>o</sub>| → ∞. The nondegeneracy condition only concerns a specific set of zeros of ƒ and we make no assumption whatsoever on the nonconstant equilibria. The set of such equilibria can be very complicated and indeed a complete understanding of this set is usually beyond reach in dimension <i>N</i> ⋝ 2. Moreover, because of the symmetries of the equation there are always continua of such equilibria. Our result shows that the assumption "<i>u</i><sub>o</sub> has compact support" is powerful enough to guarantee that, first, the equilibria that can possibly be observed in the <i>w</i>-limit set of <i>u</i> have a rather simple structure; and, second, exactly one of them is selected. Our convergence result remains valid if Δ<i>u</i> is replaced by a general elliptic operator of the form ∑<sub><i>i<sub>7</sub>j</sub>a<sub>ij</sub>u<sub>x<sub>i</sub>x<sub>j</sub></sub> with constant coefficients <i>a<sub>ij</sub></i>.</p>
Thu, 01 Jan 2015 00:00:00 GMThttps://hdl.handle.net/1959.11/183912015-01-01T00:00:00ZFree boundary models for mosquito range movement driven by climate warminghttps://hdl.handle.net/1959.11/22644Title: Free boundary models for mosquito range movement driven by climate warming
Contributor(s): Bao, Wendi; Du, Yihong; Lin, Zhigui; Zhu, Huaiping
Abstract: As vectors, mosquitoes transmit numerous mosquito-borne diseases. Among the many factors affecting the distribution and density of mosquitoes, climate change and warming have been increasingly recognized as major ones. In this paper, we make use of three diffusive logistic models with free boundary in one space dimension to explore the impact of climate warming on the movement of mosquito range. First, a general model incorporating temperature change with location and time is introduced. In order to gain insights of the model, a simplified version of the model with the change of temperature depending only on location is analyzed theoretically, for which the dynamical behavior is completely determined and presented. The general model can be modified into a more realistic one of seasonal succession type, to take into account of the seasonal changes of mosquito movements during each year, where the general model applies only for the time period of the warm seasons of the year, and during the cold season, the mosquito range is fixed and the population is assumed to be in a hibernating status. For both the general model and the seasonal succession model, our numerical simulations indicate that the long-time dynamical behavior is qualitatively similar to the simplified model, and the effect of climate warming on the movement of mosquitoes can be easily captured. Moreover, our analysis reveals that hibernating enhances the chances of survival and successful spreading of the mosquitoes, but it slows down the spreading speed.
Mon, 01 Jan 2018 00:00:00 GMThttps://hdl.handle.net/1959.11/226442018-01-01T00:00:00ZAsymptotic Spreading Speed for the Weak Competition System with a Free Boundaryhttps://hdl.handle.net/1959.11/28556Title: Asymptotic Spreading Speed for the Weak Competition System with a Free Boundary
Contributor(s): Wang, Zhiguo; Nie, Hua; Du, Yihong
Abstract: This paper is concerned with a diffusive Lotka-Volterra type competition system with a free boundary in one space dimension. Such a system may be used to describe the invasion of a new species into the habitat of a native competitor, and its long-time dynamical behavior can be described by a spreading-vanishing dichotomy. The main purpose of this paper is to determine the asymptotic spreading speed of the invading species when its spreading is successful, which involves two systems of traveling wave type equations.
Sun, 01 Sep 2019 00:00:00 GMThttps://hdl.handle.net/1959.11/285562019-09-01T00:00:00ZLogarithmic corrections in Fisher-KPP type porous medium equationshttps://hdl.handle.net/1959.11/28558Title: Logarithmic corrections in Fisher-KPP type porous medium equations
Contributor(s): Du, Yihong; Quiros, Fernando; Zhou, Maolin
Abstract: We consider the large time behavior of solutions to the porous medium equation with a Fisher–KPP type reaction term and nonnegative, compactly supported initial function in L∞(RN) \ {0}:
<br/>
(*)
ut = Δum + u − u2 in Q := RN × R+, u(·,0) = u0 in RN, (*)
<br/>
with m > 1. It is well known that the spatial support of the solution u(·, t) to this problem remains bounded for all time t > 0 (whose boundary is called the free boundary), which is a main different feature of (*) to the corresponding semilinear case m = 1. Similar to the corresponding semilinear case m = 1, it is known that there is a minimal speed c* > 0 such that for any c ≥ c*, the equation admits a wavefront solution Φc(r): For any ν ∈ SN−1, v(x, t) := Φc(x · ν − ct) solves vt = Δvm+v−v2. When m = 1, it is well known that the long-time behavior of the solution with compact initial support can be well approximated by Φc∗ (|x| − c∗t + N+2 c* log t + O(1)), and the term N+2 c* log t is known as the logarithmic correction term. When m > 1, an analogous approximation has been an open question for N ≥ 2. In this paper, we answer this question by showing that there exists a constant c# > 0 independent of the dimension N and the initial function u0, such that for all large time, any solution of (*) is well approximated by Φc∗ (|x| − c∗t + (N−1)c# log t +O(1)). This is achieved by a careful analysis of the radial case, where the initial function u0 is radially symmetric, which enables us to give a formula for c# (involving integrals of Φc* (r)), and to replace the O(1) term by C + o(1) with C a constant depending on u0. The approximation for the general non-radial case is obtained by using the radial results and simple comparison arguments. We note that in sharp contrast to the m = 1 case, when N = 1, there is no logarithmic correction term for (*).
Wed, 01 Apr 2020 00:00:00 GMThttps://hdl.handle.net/1959.11/285582020-04-01T00:00:00ZSpreading in space-time periodic media governed by a monostable equation with free boundaries, Part 2: Spreading speedhttps://hdl.handle.net/1959.11/28561Title: Spreading in space-time periodic media governed by a monostable equation with free boundaries, Part 2: Spreading speed
Contributor(s): Ding, Weiwei; Du, Yihong; Liang, Xing
Abstract: This is Part 2 of our work aimed at classifying the long-time behavior of the solution to a free boundary problem with monostable reaction term in space-time periodic media. In Part 1 (see [2]) we have established a theory on the existence and uniqueness of solutions to this free boundary problem with continuous initial functions, as well as a spreading-vanishing dichotomy. We are now able to develop the methods of Weinberger [15,16]and others [6-10]to prove the existence of asymptotic spreading speed when spreading happens, without knowing a priori the existence of the corresponding semi-wave solutions of the free boundary problem. This is a completely different approach from earlier works on the free boundary model, where the spreading speed is determined by firstly showing the existence of a corresponding semi-wave. Such a semi-wave appears difficult to obtain by the earlier approaches in the case of space-time periodic media considered in our work here.
Tue, 01 Jan 2019 00:00:00 GMThttps://hdl.handle.net/1959.11/285612019-01-01T00:00:00ZSpreading speed for a West Nile virus model with free boundaryhttps://hdl.handle.net/1959.11/28569Title: Spreading speed for a West Nile virus model with free boundary
Contributor(s): Wang, Zhiguo; Nie, Hua; Du, Yihong
Abstract: The purpose of this paper is to determine the precise asymptotic spreading speed of the virus for a West Nile virus model with free boundary, introduced recently in Lin and Zhu (J Math Biol 75:1381–1409, 2017), based on a model of Lewis et al. (Bull Math Biol 68:3–23, 2006).We show that this speed is uniquely defined by a semiwave solution associated with theWest Nile virus model. To find such a semiwave solution, we firstly consider a general cooperative system over the half-line [0,∞), and prove the existence of amonotone solution by an upper and lower solution approach; we then establish the existence and uniqueness of the desired semiwave solution by applying this method together with some other techniques including the sliding method. Our result indicates that the asymptotic spreading speed of theWest Nile virus model with free boundary is strictly less than that of the corresponding model in Lewis et al. (2006).
Tue, 01 Jan 2019 00:00:00 GMThttps://hdl.handle.net/1959.11/285692019-01-01T00:00:00ZFinite Morse index solutions of weighted elliptic equations and the critical exponentshttps://hdl.handle.net/1959.11/18388Title: Finite Morse index solutions of weighted elliptic equations and the critical exponents
Contributor(s): Du, Yihong; Guo, Zongming
Abstract: We study the behavior of finite Morse index solutions to the weighted elliptic equation...
Thu, 01 Jan 2015 00:00:00 GMThttps://hdl.handle.net/1959.11/183882015-01-01T00:00:00ZBoundary behavior of positive solutions to nonlinear elliptic equations with Hardy potentialhttps://hdl.handle.net/1959.11/18392Title: Boundary behavior of positive solutions to nonlinear elliptic equations with Hardy potential
Contributor(s): Du, Yihong; Wei, Lei
Abstract: In this paper, we study the boundary behavior of positive solutions of the following equation...
Thu, 01 Jan 2015 00:00:00 GMThttps://hdl.handle.net/1959.11/183922015-01-01T00:00:00ZMultiple steady-states in phytoplankton population induced by photoinhibitionhttps://hdl.handle.net/1959.11/18393Title: Multiple steady-states in phytoplankton population induced by photoinhibition
Contributor(s): Du, Yihong; Hsu, Sze-Bi; Lou, Yuan
Abstract: We study the effect of photoinhibition in a nonlocal reaction-diffusion-advection equation, which models the dynamics of a single phytoplankton species in a water column where the growth of the species depends solely on light. Our results show that, in contrast to the case of no photoinhibition, where at most one positive steady state can exist, the model with photoinhibition possesses at least two positive steady states in certain parameter ranges. Our approach involves bifurcation theory and perturbation-reduction arguments.
Thu, 01 Jan 2015 00:00:00 GMThttps://hdl.handle.net/1959.11/183932015-01-01T00:00:00ZSpreading speed and profile for nonlinear Stefan problems in high space dimensionshttps://hdl.handle.net/1959.11/18394Title: Spreading speed and profile for nonlinear Stefan problems in high space dimensions
Contributor(s): Du, Yihong; Matsuzawa, Hiroshi; Zhou, Maolin
Abstract: We consider nonlinear diffusion problems of the form ut = Δu + f(u) with Stefan type free boundary conditions, where the nonlinear term f(u) is of monostable, bistable or combustion type. Such problems are used as an alternative model (to the corresponding Cauchy problem) to describe the spreading of a biological or chemical species, where the free boundary represents the expanding front. We are interested in its long-time spreading behavior which, by recent results of Du, Matano and Wang [10], is largely determined by radially symmetric solutions. Therefore we will examine the radially symmetric case, where the equation is satisfied in |x| < h(t), with |x| = h(t) the free boundary. We assume that spreading happens, namely limt→∞h(t) = ∞, limt→∞u(t,|x|) =1. For the case of one space dimension (N = 1), Du and Lou [8]proved that limt→∞h(t)t=c∗ for some c∗ > 0. Subsequently, sharper estimate of the spreading speed was obtained by the authors of the current paper in [11], in the form that limt→∞[h(t) − c∗t] =ˆ Ĥ∈R¹. In this paper, we consider the case N ≥ 2 and show that a logarithmic shifting occurs, namely there exists c͙ > 0 independent of N such that limt→∞[h(t) − c∗t + (N − 1)c͙logt] =ˆ ĥ∈R¹. At the same time, we also obtain a rather clear description of the spreading profile of u(t,r). These results reveal striking differences from the spreading behavior modeled by the corresponding Cauchy problem.
Thu, 01 Jan 2015 00:00:00 GMThttps://hdl.handle.net/1959.11/183942015-01-01T00:00:00ZPulsating semi-waves in periodic media and spreading speed determined by a free boundary modelhttps://hdl.handle.net/1959.11/18396Title: Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model
Contributor(s): Du, Yihong; Liang, Xing
Abstract: We consider a radially symmetric free boundary problem with logistic nonlinear term. The spatial environment is assumed to be asymptotically periodic at infinity in the radial direction. For such a free boundary problem, it is known from [7] that a spreading-vanishing dichotomy holds. However, when spreading occurs, only upper and lower bounds are obtained in [7] for the asymptotic spreading speed. In this paper, we investigate one-dimensional pulsating semi-waves in spatially periodic media. We prove existence, uniqueness of such pulsating semi-waves, and show that the asymptotic spreading speed of the free boundary problem coincides with the speed of the corresponding pulsating semi-wave.
Thu, 01 Jan 2015 00:00:00 GMThttps://hdl.handle.net/1959.11/183962015-01-01T00:00:00ZThe Fisher-KPP equation over simple graphs: varied persistence states in river networkshttps://hdl.handle.net/1959.11/28555Title: The Fisher-KPP equation over simple graphs: varied persistence states in river networks
Contributor(s): Du, Yihong; Lou, Bendong; Peng, Rui; Zhou, Maolin
Abstract: In this article, we study the dynamical behaviour of a new species spreading from a location in a river network where two or three branches meet, based on the widely used Fisher-KPP advection-diffusion equation. This local river system is represented by some simple graphs with every edge a half infinite line, meeting at a single vertex. We obtain a rather complete description of the long-time dynamical behaviour for every case under consideration, which can be classified into three different types (called a trichotomy), according to the water flow speeds in the river branches, which depend crucially on the topological structure of the graph representing the local river system and on the cross section areas of the branches. The trichotomy includes two different kinds of persistence states, and the state called "persistence below carrying capacity" here appears new.
Wed, 01 Apr 2020 00:00:00 GMThttps://hdl.handle.net/1959.11/285552020-04-01T00:00:00ZNumerical studies of a class of reaction-diffusion equations with Stefan conditionshttps://hdl.handle.net/1959.11/28559Title: Numerical studies of a class of reaction-diffusion equations with Stefan conditions
Contributor(s): Liu, Shuang; Du, Yihong; Liu, Xinfeng
Abstract: It is always very difficult to efficiently and accurately solve a system of differential equations coupled with moving free boundaries, while such a system has been widely applied to describe many physical/biological phenomena such as the dynamics of spreading population. The main purpose of this paper is to introduce efficient numerical methods within a general framework for solving such systems with moving free boundaries. The major numerical challenge is to track the moving free boundaries, especially for high spatial dimensions. To overcome this, a front tracking framework coupled with implicit solver is first introduced for the 2D model with radial symmetry. For the general 2D model, a level set approach is employed to more efficiently treat complicated topological changes. The accuracy and order of convergence for the proposed methods are discussed, and the numerical simulations agree well with theoretical results.
Wed, 01 Jan 2020 00:00:00 GMThttps://hdl.handle.net/1959.11/285592020-01-01T00:00:00ZRevisiting the Fisher-Kolmogorov-Petrovsky-Piskunov equation to interpret the spreading-extinction dichotomyhttps://hdl.handle.net/1959.11/28560Title: Revisiting the Fisher-Kolmogorov-Petrovsky-Piskunov equation to interpret the spreading-extinction dichotomy
Contributor(s): El-Hachem, Maud; McCue, Scott W; Jin, Wang; Du, Yihong; Simpson, Matthew J
Abstract: The Fisher-Kolmogorov-Petrovsky-Piskunov model, also known as the Fisher-KPP model, supports travelling wave solutions that are successfully used to model numerous invasive phenomena with applications in biology, ecology and combustion theory. However, there are certain phenomena that the Fisher-KPP model cannot replicate, such as the extinction of invasive populations. The Fisher-Stefan model is an adaptation of the Fisher-KPP model to include a moving boundary whose evolution is governed by a Stefan condition. The Fisher-Stefan model also supports travelling wave solutions; however, a key additional feature of the Fisher-Stefan model is that it is able to simulate population extinction, giving rise to a spreading-extinction dichotomy. In this work, we revisit travelling wave solutions of the Fisher-KPP model and show that these results provide new insight into travelling wave solutions of the Fisher-Stefan model and the spreading-extinction dichotomy. Using a combination of phase plane analysis, perturbation analysis and linearization, we establish a concrete relationship between travelling wave solutions of the Fisher-Stefan model and often-neglected travelling wave solutions of the Fisher-KPP model. Furthermore, we give closed-form approximate expressions for the shape of the travelling wave solutions of the Fisher-Stefan model in the limit of slow travelling wave speeds, c≪1.
Sun, 01 Sep 2019 00:00:00 GMThttps://hdl.handle.net/1959.11/285602019-09-01T00:00:00ZThe dynamics of a Fisher-KPP nonlocal diffusion model with free boundarieshttps://hdl.handle.net/1959.11/28570Title: The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries
Contributor(s): Cao, Jia-Feng; Du, Yihong; Li, Fang; Li, Wan-Tong
Abstract: We introduce and study a class of free boundary models with "nonlocal diffusion", which are natural extensions of the free boundary models in [16]and elsewhere, where “local diffusion” is used to describe the population dispersal, with the free boundary representing the spreading front of the species. We show that this nonlocal problem has a unique solution defined for all time, and then examine its long-time dynamical behavior when the growth function is of Fisher-KPP type. We prove that a spreading-vanishing dichotomy holds, though for the spreading-vanishing criteria significant differences arise from the well known local diffusion model in [16].
Tue, 01 Jan 2019 00:00:00 GMThttps://hdl.handle.net/1959.11/285702019-01-01T00:00:00ZThe Work of Norman Dancerhttps://hdl.handle.net/1959.11/28571Title: The Work of Norman Dancer
Contributor(s): Du, Yihong
Abstract: 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.
Fri, 01 Nov 2019 00:00:00 GMThttps://hdl.handle.net/1959.11/285712019-11-01T00:00:00ZSpreading-vanishing dichotomy in a diffusive logistic model with a free boundary, IIhttps://hdl.handle.net/1959.11/9936Title: Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, II
Contributor(s): Du, Yihong; Guo, Zongming
Abstract: We study the diffusive logistic equation with a free boundary in higher space dimensions and heterogeneous environment. Such a model may be used to describe the spreading of a new or invasive species, with the free boundary representing the expanding front. For simplicity, we assume that the environment and the solution are radially symmetric. In the special case of one space dimension and homogeneous environment, this free boundary problem was investigated in Du and Lin (2010) [10]. We prove that the spreading-vanishing dichotomy established in Du and Lin (2010) [10] still holds in the more general and ecologically realistic setting considered here. Moreover, when spreading occurs, we obtain best possible upper and lower bounds for the spreading speed of the expanding front. When the environment is asymptotically homogeneous at infinity, these two bounds coincide. Our results indicate that the asymptotic spreading speed determined by this model does not depend on the spatial dimension.
Sat, 01 Jan 2011 00:00:00 GMThttps://hdl.handle.net/1959.11/99362011-01-01T00:00:00ZOrder Structure and Topological Methods in Nonlinear Partial Differential Equations: Vol 1: Maximum Principles and Applicationshttps://hdl.handle.net/1959.11/2471Title: Order Structure and Topological Methods in Nonlinear Partial Differential Equations: Vol 1: Maximum Principles and Applications
Contributor(s): Du, Yihong
Abstract: This is volume one of a two volume series. The intention is to provide a reference book for researchers in nonlinear partial differential equations and nonlinear functional analysis, especially for postgraduate students who want to be led to some of the current research topics. It could be used as a textbook for postgraduate students, either in formal classes or in working seminars. In these two volumes, we attempt to use order structure as a thread to introduce the various versions of the maximum principles, the fixed point index theory, and the relevant part of critical point theory and conley index theory. The emphasize is on their applications, and we try to demonstrate the usefulness of these tools by choosing applications to problems in partial differential equations that are of considerable concern of current research.
Sun, 01 Jan 2006 00:00:00 GMThttps://hdl.handle.net/1959.11/24712006-01-01T00:00:00ZEffects of environmental heterogeneity on species spreading via numerical analysis of some free boundary modelshttps://hdl.handle.net/1959.11/52931Title: Effects of environmental heterogeneity on species spreading via numerical analysis of some free boundary models
Contributor(s): Kamruzzaman, Khan; Schaerf, Timothy M; Du, Yihong
Abstract: <p>This paper investigates the effect of environmental heterogeneity on species spreading via numerical simulation of suitable reaction-diffusion models with free boundaries. We focus on the changes of long-time dynamics (establishment or extinction) and spreading speeds of the species as the parameters describing the heterogeneity of the environment are varied. For the single species model in time-periodic environment and in space-periodic environment theoretically treated in [15,16], we obtain more detailed properties here. Among other results, our numerical simulation suggests that, in a time-periodic or space-periodic environment, moderate increase of the oscillation scale enhances the chances of establishment as well as the spreading speed of the species. We also numerically examine a related model with two competing species, which was treated in [34,28,24] recently and reduces to the single species free boundary model when one of the species is absent. Our numerical results, obtained by varying the parameters in the time-periodic and space-periodic terms of the model, suggest that heterogeneity of the environment enhances the invasion of the two species (as in the single species model), although there are subtle differences of the influences felt by the two. Some intriguing phenomena revealed in our simulations suggest that heterogeneity of the environment decreases the level of predictability of the competition outcome.</p>
https://hdl.handle.net/1959.11/52931Regularity and Asymptotic Behavior of Nonlinear Stefan Problemshttps://hdl.handle.net/1959.11/17044Title: Regularity and Asymptotic Behavior of Nonlinear Stefan Problems
Contributor(s): Du, Yihong; Matano, Hiroshi; Wang, Kelei
Abstract: We study the following nonlinear Stefan problem 'ut−dΔu=g(u)u=0andut=μ|∇xu|2u(0,x)=u0(x)forx∈Ω(t),t>0,forx∈ (t),t>0,forx∈Ω0', where Ω(t)⊂Rn ( n≧2 ) is bounded by the free boundary Γ(t) , with Ω(0)=Ω0 , μ and d are given positive constants. The initial function u 0 is positive in Ω0 and vanishes on ∂Ω0 . The class of nonlinear functions g(u) includes the standard monostable, bistable and combustion type nonlinearities. We show that the free boundary Γ(t) is smooth outside the closed convex hull of Ω0 , and as t→∞ , either Ω(t) expands to the entire Rn , or it stays bounded. Moreover, in the former case, Γ(t) converges to the unit sphere when normalized, and in the latter case, u→0 uniformly. When g(u)=au−bu2, we further prove that in the case Ω(t) expands to Rn , u→a/b as t→∞ , and the spreading speed of the free boundary converges to a positive constant; moreover, there exists μ∗≧0 such that Ω(t) expands to Rn exactly when μ>μ∗.
Wed, 01 Jan 2014 00:00:00 GMThttps://hdl.handle.net/1959.11/170442014-01-01T00:00:00Z