scholarly journals The dynamics of nonlocal diffusion systems with different free boundaries

2020 ◽  
Vol 19 (7) ◽  
pp. 3651-3672
Author(s):  
Lei Li ◽  
◽  
Jianping Wang ◽  
Mingxin Wang
Keyword(s):  
Nonlocal Diffusion ◽  
Free Boundaries ◽  
Diffusion Systems

scholarly journals Two species nonlocal diffusion systems with free boundaries

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yihong Du ◽  
Mingxin Wang ◽  
Meng Zhao
Keyword(s):  
Free Boundary ◽  
Free Boundary Problems ◽  
Reaction Diffusion ◽  
Single Species ◽  
Diffusion Problem ◽  
Nonlocal Diffusion ◽  
Free Boundaries ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Local Diffusion

<p style='text-indent:20px;'>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 [<xref ref-type="bibr" rid="b7">7</xref>], 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>

scholarly journals The dynamics of a degenerate epidemic model with nonlocal diffusion and free boundaries

2020 ◽  
Vol 269 (4) ◽  
pp. 3347-3386 ◽  
Author(s):  
Meng Zhao ◽  
Yang Zhang ◽  
Wan-Tong Li ◽  
Yihong Du
Keyword(s):  
Epidemic Model ◽  
Nonlocal Diffusion ◽  
Free Boundaries

scholarly journals Long-time dynamics of an epidemic model with nonlocal diffusion and free boundaries

2022 ◽  
Vol 30 (1) ◽  
pp. 289-313
Author(s):  
Ting-Ying Chang ◽  
◽  
Yihong Du
Keyword(s):  
Epidemic Model ◽  
Reaction Diffusion ◽  
Host Population ◽  
Kernel Functions ◽  
Nonlocal Diffusion ◽  
Threshold Condition ◽  
Free Boundaries ◽  
Time Dynamics ◽  
Long Time ◽  
Long Time Dynamics

<abstract><p>In this paper, we consider a reaction-diffusion epidemic model with nonlocal diffusion and free boundaries, which generalises the free-boundary epidemic model by Zhao et al. <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup> by including spatial mobility of the infective host population. We obtain a rather complete description of the long-time dynamics of the model. For the reproduction number $ R_0 $ arising from the corresponding ODE model, we establish its relationship to the spreading-vanishing dichotomy via an associated eigenvalue problem. If $ R_0 \le 1 $, we prove that the epidemic vanishes eventually. On the other hand, if $ R_0 &gt; 1 $, we show that either spreading or vanishing may occur depending on its initial size. In the case of spreading, we make use of recent general results by Du and Ni <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup> to show that finite speed or accelerated spreading occurs depending on whether a threshold condition is satisfied by the kernel functions in the nonlocal diffusion operators. In particular, the rate of accelerated spreading is determined for a general class of kernel functions. Our results indicate that, with all other factors fixed, the chance of successful spreading of the disease is increased when the mobility of the infective host is decreased, reaching a maximum when such mobility is 0 (which is the situation considered by Zhao et al. <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>).</p></abstract>

scholarly journals Time‐scale analysis nonlocal diffusion systems, applied to disease models

2020 ◽  
Vol 43 (15) ◽  
pp. 8632-8643
Author(s):  
Marcone Pereira ◽  
Sergio Oliva ◽  
Larissa Sartori
Keyword(s):  
Time Scale ◽  
Nonlocal Diffusion ◽  
Disease Models ◽  
Scale Analysis ◽  
Diffusion Systems
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