Presentation Information
[MS09-03]Spreading of species modeled by a reaction-diffusion equation with free boundaries
*Yuki Kaneko1 (1. Kanto Gakuin University (Japan))
Keywords:
spreading and vanishing,free boundary problem,reaction-diffusion equation,large-time behavior,propagating terrace
Since the pioneering work of Skellam in 1951, the spreading of invasive species is one of the most important topics in mathematical biology. Traveling wave solutions to reaction-diffusion equations have been used to understand invasion phenomena. In 2010, Du and Lin [2] proposed a new mathematical model in which the spreading front of biological species was explicitly shown as a free boundary. They proved a spreading- vanishing dichotomy which says, as time tends to infinity, the free boundary goes to infinity and a density function converges to a positive constant state (called spreading), or the free boundary stays in a finite interval and the density function converges to 0 (called vanishing). See more information in [1, 3].
We discuss a related free boundary problem
ut −duxx =f(u), t>0, g(t)<x<h(t),
where g(t) and h(t) are determined by one-phase Stefan conditions
g′(t) = −μ1ux(t,g(t)), h′(t) = −μ2ux(t,h(t))
with u(t,g(t)) = 0, u(t,h(t)) = 0 and d, μ1, μ2 > 0. In this problem, u(t,x) denotes the population density of the species and (g(t),h(t)) represents the habitat of the species. We assume that f(u) is subject to a positive bistable term which was used to model the population dynamics of spruce budworm in [5].
We are interested in the asymptotic behavior of the spreading fronts g(t),h(t) and the eventual distribution of u(t,x) as t → ∞. Then we can classify solutions into four cases: big spreading, small spreading, transition and vanishing, and show that a big spreading solution forms a propagating terrace similar to the one of [4]. The latter result means a two-stage spreading process. In other words, the species stays at the low density level for a while, and later large population comes up at a speed slower than that of the spreading front.
References
[1] G. Bunting, Y. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583–603.
[2] Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377–405.
[3] Y. Kaneko and Y. Yamada, A free boundary problem for a reaction-diffusion equation appearing in ecology, Adv. Math. Sci. Appl., 21 (2011), 467–492.
[4] Y. Kaneko, H. Matsuzawa and Y. Yamada, Asymptotic profiles of solutions and propagating terrace for a free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity, SIAM J. Math. Anal., 52 (2020), 65–103.
[5] D. Ludwig, D. G. Aronson, H. F. Weinberger, Spatial patterning of the space budworm, J. Math. Biol., 8 (1979) 217–258.
We discuss a related free boundary problem
ut −duxx =f(u), t>0, g(t)<x<h(t),
where g(t) and h(t) are determined by one-phase Stefan conditions
g′(t) = −μ1ux(t,g(t)), h′(t) = −μ2ux(t,h(t))
with u(t,g(t)) = 0, u(t,h(t)) = 0 and d, μ1, μ2 > 0. In this problem, u(t,x) denotes the population density of the species and (g(t),h(t)) represents the habitat of the species. We assume that f(u) is subject to a positive bistable term which was used to model the population dynamics of spruce budworm in [5].
We are interested in the asymptotic behavior of the spreading fronts g(t),h(t) and the eventual distribution of u(t,x) as t → ∞. Then we can classify solutions into four cases: big spreading, small spreading, transition and vanishing, and show that a big spreading solution forms a propagating terrace similar to the one of [4]. The latter result means a two-stage spreading process. In other words, the species stays at the low density level for a while, and later large population comes up at a speed slower than that of the spreading front.
References
[1] G. Bunting, Y. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583–603.
[2] Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377–405.
[3] Y. Kaneko and Y. Yamada, A free boundary problem for a reaction-diffusion equation appearing in ecology, Adv. Math. Sci. Appl., 21 (2011), 467–492.
[4] Y. Kaneko, H. Matsuzawa and Y. Yamada, Asymptotic profiles of solutions and propagating terrace for a free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity, SIAM J. Math. Anal., 52 (2020), 65–103.
[5] D. Ludwig, D. G. Aronson, H. F. Weinberger, Spatial patterning of the space budworm, J. Math. Biol., 8 (1979) 217–258.