Unbounded Habitat Research Articles

Wave speed and critical patch size for integro-difference equations with a strong Allee effect.

Simplified conditions are given for the existence and positivity of wave speed for an integro-difference equation with a strong Allee effect and an unbounded habitat. The results are used to obtain the existence of a critical patch size for an equation with a bounded habitat. It is shown that if the wave speed is positive there exists a critical patch size such that for a habitat size above the critical patch size solutions can persist in space, and if the wave speed is negative solutions always approach zero. An analytical integral formula is developed to determine the critical patch size when the Laplace dispersal kernel is used, and this formula shows existence of multiple equilibrium solutions. Numerical simulations are provided to demonstrate connections among the wave speed, critical patch size, and Allee threshold.

Dynamics of a time-periodic and delayed reaction-diffusion model with a quiescent stage

In this paper, we study a time-periodic and delayed reaction-diffusion system with quiescent stage in both unbounded and bounded habitat domains. In unbounded habitat domain R, we first prove the existence of the asymptotic spreading speed and then show that it coincides with the minimal wave speed for monotone periodic traveling waves. In a bounded habitat domain Omega subset of R-N (N >= 1), we obtain the threshold result on the global attractivity of either the zero solution or the unique positive time-periodic solution of the system.

Clines with partial panmixia

In spatially distributed populations, global panmixia can be regarded as the limiting case of long-distance migration. The effect of incorporating partial panmixia into single-locus clines maintained by migration and selection is investigated. In a diallelic, two-deme model without dominance, partial panmixia can increase or decrease both the polymorphic area in the plane of the migration rates and the equilibrium gene-frequency difference between the two demes. For multiple alleles, under the assumptions that the number of demes is large and both migration and selection are arbitrary but weak, a system of integro-partial differential equations is derived. For two alleles with conservative migration, (i) a Lyapunov functional is found, suggesting generic global convergence of the gene frequency; (ii) conditions for the stability or instability of the fixation states, and hence for a protected polymorphism, are obtained; and (iii) a variational representation of the minimal selection-migration ratio λ 0 (the principal eigenvalue of the linearized system) for protection from loss is used to prove that λ 0 is an increasing function of the panmictic rate and to deduce the effect on λ 0 of changes in selection and migration. The unidimensional step-environment with uniform population density, homogeneous, isotropic migration, and no dominance is examined in detail: An explicit characteristic equation is derived for λ 0 ; bounds on λ 0 are established; and λ 0 is approximated in four limiting cases. An explicit formula is also deduced for the globally asymptotically stable cline in an unbounded habitat with a symmetric environment; partial panmixia maintains some polymorphism even as the distance from the center of the cline tends to infinity.