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1.
Arnaud Ducrot 《Journal of Differential Equations》2011,250(1):410-449
This work is devoted to the study of travelling wave solutions for some size structured model in population dynamics. The population under consideration is also spatially structured and has a nonlocal spatial reproduction. This phenomenon may model the invasion of plants within some empty landscape. Since the corresponding unspatially structured size structured models may induce oscillating dynamics due to Hopf bifurcations, the aim of this work is to prove the existence of point to sustained oscillating solution travelling waves for the spatially structured problem. From a biological view point, such solutions represent the spatial invasion of some species with spatio-temporal patterns at the place where the population is established. 相似文献
2.
Rui Peng 《Journal of Differential Equations》2009,247(4):1096-1119
To capture the impact of spatial heterogeneity of environment and movement of individuals on the persistence and extinction of a disease, Allen et al. in [L.J.S. Allen, B.M. Bolker, Y. Lou, A.L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst. Ser. A 21 (1) (2008) 1-20] proposed a spatial SIS (susceptible-infected-susceptible) reaction-diffusion model, and studied the existence, uniqueness and particularly the asymptotic behavior of the endemic equilibrium as the diffusion rate of the susceptible individuals goes to zero in the case where a so-called low-risk subhabitat is created. In this work, we shall provide further understanding of the impacts of large and small diffusion rates of the susceptible and infected population on the persistence and extinction of the disease, which leads us to determine the asymptotic behaviors of the endemic equilibrium when the diffusion rate of either the susceptible or infected population approaches to infinity or zero in the remaining cases. Consequently, our results reveal that, in order to eliminate the infected population at least in low-risk area, it is necessary that one will have to create a low-risk subhabitat and reduce at least one of the diffusion rates to zero. In this case, our results also show that different strategies of controlling the diffusion rates of individuals may lead to very different spatial distributions of the population; moreover, once the spatial environment is modified to include a low-risk subhabitat, the optimal strategy of eradicating the epidemic disease is to restrict the diffusion rate of the susceptible individuals rather than that of the infected ones. 相似文献
3.
Elena Trofimchuk 《Journal of Differential Equations》2009,246(4):1422-1787
The aim of this paper is to study the existence and the geometry of positive bounded wave solutions to a non-local delayed reaction-diffusion equation of the monostable type. 相似文献
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Guo-Bao Zhang Wan-Tong Li Yu-Juan Sun 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(12):4466-4474
This paper is concerned with the existence and asymptotic behavior of solutions of a nonlocal dispersal equation. By means of super-subsolution method and monotone iteration, we first study the existence and asymptotic behavior of solutions for a general nonlocal dispersal equation. Then, we apply these results to our equation and show that the nonnegative solution is unique, and the behavior of this solution depends on parameter λ in equation. For λ≤λ1(Ω), the solution decays to zero as t→∞; while for λ>λ1(Ω), the solution converges to the unique positive stationary solution as t→∞. In addition, we show that the solution blows up under some conditions. 相似文献
6.
Guo-Bao Zhang 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(15):5030-5047
This paper is concerned with the traveling waves in a single species population model which is derived by considering the nonlocal dispersal and age-structure. If the birth function is monotone, then the existence of traveling wavefront is reduced to the existence of a pair of super and subsolutions without the requirement of smoothness. It is proved that the traveling wavefront is strictly increasing and unique up to a translation. The asymptotic behavior of traveling wavefronts is also obtained. If the birth function is not monotone, the existence of traveling wave solution is affirmed by introducing two auxiliary nonlocal dispersal equations with quasi-monotonicity. 相似文献
7.
The semilinear parabolic system that describes the evolution of the gene frequencies in the diffusion approximation for migration and selection at a multiallelic locus is investigated. The population occupies a finite habitat of arbitrary dimensionality and shape (i.e., a bounded, open domain in Rd). The selection coefficients depend on position and may depend on the gene frequencies; the drift and diffusion coefficients may depend on position. Sufficient conditions are given for the global loss of an allele and for its protection from loss. A sufficient condition for the existence of at least one internal equilibrium is also offered, and the profile of any internal equilibrium in the zero-migration limit is obtained. 相似文献
8.
Jia-Fang ZhangWan-Tong Li Yu-Xia Wang 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(3):847-858
The main purpose of this work is to investigate the effects of cross-diffusion in a strongly coupled predator-prey system. By a linear stability analysis we find the conditions which allow a homogeneous steady state (stable for the kinetics) to become unstable through a Turing mechanism. In particular, it is shown that Turing instability of the reaction-diffusion system can disappear due to the presence of the cross-diffusion, which implies that the cross-diffusion induced stability can be regarded as the cross-stability of the corresponding reaction-diffusion system. Furthermore, we consider the existence and non-existence results concerning non-constant positive steady states (patterns) of the system. We demonstrate that cross-diffusion can create non-constant positive steady-state solutions. These results exhibit interesting and very different roles of the cross-diffusion in the formation and the disappearance of the Turing instability. 相似文献
9.
Pedro Marín-Rubio José Real 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(5):2012-2030
We prove the existence of solutions for a Navier-Stokes model in two dimensions with an external force containing infinite delay effects in the weighted space Cγ(H). Then, under additional suitable assumptions, we prove the existence and uniqueness of a stationary solution and the exponential decay of the solutions of the evolutionary problem to this stationary solution. Finally, we study the existence of pullback attractors for the dynamical system associated to the problem under more general assumptions. 相似文献
10.
Adrian Gomez 《Journal of Differential Equations》2011,250(4):1767-1787
In the early 2000's, Gourley (2000), Wu et al. (2001), Ashwin et al. (2002) initiated the study of the positive wavefronts in the delayed Kolmogorov-Petrovskii-Piskunov-Fisher equation
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11.
This is the second part of a series of study on the stability of traveling wavefronts of reaction-diffusion equations with time delays. In this paper we will consider a nonlocal time-delayed reaction-diffusion equation. When the initial perturbation around the traveling wave decays exponentially as x→−∞ (but the initial perturbation can be arbitrarily large in other locations), we prove the asymptotic stability of all traveling waves for the reaction-diffusion equation, including even the slower waves whose speed are close to the critical speed. This essentially improves the previous stability results by Mei and So [M. Mei, J.W.-H. So, Stability of strong traveling waves for a nonlocal time-delayed reaction-diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008) 551-568] for the speed with a small initial perturbation. The approach we use here is the weighted energy method, but the weight function is more tricky to construct due to the property of the critical wavefront, and the difficulty arising from the nonlocal nonlinearity is also overcome. Finally, by using the Crank-Nicholson scheme, we present some numerical results which confirm our theoretical study. 相似文献
12.
In this paper, we study a class of time-delayed reaction-diffusion equation with local nonlinearity for the birth rate. For all wavefronts with the speed c>c∗, where c∗>0 is the critical wave speed, we prove that these wavefronts are asymptotically stable, when the initial perturbation around the traveling waves decays exponentially as x→−∞, but the initial perturbation can be arbitrarily large in other locations. This essentially improves the stability results obtained by Mei, So, Li and Shen [M. Mei, J.W.-H. So, M.Y. Li, S.S.P. Shen, Asymptotic stability of traveling waves for the Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 579-594] for the speed with small initial perturbation and by Lin and Mei [C.-K. Lin, M. Mei, On travelling wavefronts of the Nicholson's blowflies equations with diffusion, submitted for publication] for c>c∗ with sufficiently small delay time r≈0. The approach adopted in this paper is the technical weighted energy method used in [M. Mei, J.W.-H. So, M.Y. Li, S.S.P. Shen, Asymptotic stability of traveling waves for the Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 579-594], but inspired by Gourley [S.A. Gourley, Linear stability of travelling fronts in an age-structured reaction-diffusion population model, Quart. J. Mech. Appl. Math. 58 (2005) 257-268] and based on the property of the critical wavefronts, the weight function is carefully selected and it plays a key role in proving the stability for any c>c∗ and for an arbitrary time-delay r>0. 相似文献
13.
A reaction-diffusion population model with a general time-delayed growth rate per capita is considered. The growth rate per capita can be logistic or weak Allee effect type. From a careful analysis of the characteristic equation, the stability of the positive steady state solution and the existence of forward Hopf bifurcation from the positive steady state solution are obtained via the implicit function theorem, where the time delay is used as the bifurcation parameter. The general results are applied to a “food-limited” population model with diffusion and delay effects as well as a weak Allee effect population model. 相似文献
14.
We develop a perturbation argument based on existing results on asymptotic autonomous systems and the Fredholm alternative theory that yields the persistence of traveling wavefronts for reaction-diffusion equations with nonlocal and delayed nonlinearities, when the time lag is relatively small. This persistence result holds when the nonlinearity of the corresponding ordinary reaction-diffusion system is either monostable or bistable. We then illustrate this general result using five different models from population biology, epidemiology and bio-reactors. 相似文献
15.
We show that the solutions of a population equation with age structure and delayed birth process have asynchronous exponential growth (AEG). We use operator matrices and Hille-Yosida operators as well as Perron-Frobenius techniques. 相似文献
16.
《Quaestiones Mathematicae》2013,36(8):1073-1082
AbstractIn this paper we study a two-phase population model, which distinguishes the population by two different stagesBy the standard technique of characteristics, this population equation is transformed as the ordinary differential equation with nonautonomous pastwhere 1 ≤ p < ∞ and I = [?r, 0] (finite delay) or I = (?∞, 0] (infinite delay), E a Banach space, Φ : W1,p(I, E) → E a linear delay operator and B a nonlinear operator on E. The main result of this paper is the well-posedness of this delay equation by using the (right) multiplicative perturbation result of Desch and Schappacher in [8]. 相似文献
17.
Shangbing Ai 《Journal of Differential Equations》2007,232(1):104-133
We study the existence of traveling wave fronts for a reaction-diffusion equation with spatio-temporal delays and small parameters. The equation reduces to a generalized Fisher equation if small parameters are zero. We present two results. In the first one, we deal with the equation with very general kernels and show the persistence of Fisher wave fronts for all sufficiently small parameters. In the second one, we deal with some particular kernels, with which the nonlocal equation can be reduced to a system of singularly perturbed ODEs, and we are then able to apply the geometric singular perturbation theory and phase plane arguments to this system to show the existence of the minimal wave speed, the existence of a continuum of wave fronts, and the global uniqueness of the physical wave front with each wave speed. 相似文献
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19.
In this paper, we consider travelling wave solutions for the diffusive Nicholson’s blowflies equation incorporating time delay and diffusion. Special attention is paid to the modelling of the time delay to incorporate associated non-local spatial terms which account for the drift of individuals to their present position from their possible positions at previous times. For the strong generic delay kernel, we show that travelling wave solutions exist provided that the delay is sufficiently small, using the geometric singular perturbation theory. 相似文献
20.
Robert Stephen Cantrell Chris Cosner Yuan Lou 《Journal of Differential Equations》2008,245(12):3687-3703
We consider reaction-diffusion-advection models for spatially distributed populations that have a tendency to disperse up the gradient of fitness, where fitness is defined as a logistic local population growth rate. We show that in temporally constant but spatially varying environments such populations have equilibrium distributions that can approximate those that would be predicted by a version of the ideal free distribution incorporating population dynamics. The modeling approach shows that a dispersal mechanism based on local information about the environment and population density can approximate the ideal free distribution. The analysis suggests that such a dispersal mechanism may sometimes be advantageous because it allows populations to approximately track resource availability. The models are quasilinear parabolic equations with nonlinear boundary conditions. 相似文献