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An integrodifference model that describes the spread of invading species on a periodically fragmented environment is analyzed to derive an asymptotic speed of range expansion. We consider the case that the redistribution kernel is given by an exponentially damping function and the population growth is subject to a Ricker function in which the intrinsic growth rate is specified by a spatially periodic step-function. We first derive a condition for successful invasion of a small propagule, and then provide a mathematical formula for the rate of spread. Based on the speeds calculated from the formula for various combinations of parameter values, we discuss how the habitat fragmentation influences the invasion speed. The speeds are also compared with the corresponding speeds when the dispersal kernel is replaced by a Gaussian.
This paper considers the dynamics of a general nonlinear structured population model governed by ordinary differential equations. We are especially concerned with the survival possibility of structured populations. Our results show that, under a certain mild condition, the instability of the population free equilibrium point implies that the structured population survives in the sense of permanence. Furthermore, the relationship between the basic reproduction number and the instability of the population free equilibrium point provides simple criteria for population survival. The results are applied to both stage-structured and spatially structured models.
The source-sink dynamics is a major hypothesis to explain dispersal-mediated coexistence of locally exclusive competitors. We study Lotka--Volterra diffusive models of indirect competition in patchy metacommunities. In a model of exploitative competition, we numerically show that the effect of resource movement on the coexistence depends on demographic factors that create source-sink structures and that the dispersal rate of the superior competitor need not be higher than that of the inferior to promote dispersal-mediated coexistence. In a model of apparent competition, we analytically prove that dispersal can make coexistence possible even if any patches are sinks for the inferior resource species. The requirement for this coexistence is the lower dispersal rate of the inferior competitor. We conclude that dispersal among patches can be a mechanism to save inferior indirect competitors from regional extinction and that the level of spatial heterogeneity need not be so high to reverse the competitive rankings among patches.
Sufficient conditions for permanence of a general periodic single-species system with periodic impulsive perturbations are obtained via comparison theory of impulsive differential equations. An application is given to the periodic impulsive logistic system.
It is known that Turing systems in two dimensions produce spotted, striped, and labyrinthine patterns. In three dimensions, a greater variety of patterns is possible. By numerical simulation of the FitzHugh--Nagumo type of reaction-diffusion system, we have obtained not only lamellar, hexagonal and spherical structures (BCC and FCC) but also gyroid, Fddd, and perforated lamellar structures. The domains of these three structures constitute interconnected regular networks, a characteristic occurring in three dimensions. Moreover, we derive the Lyapunov functional by reducing the system, and we evaluate this functional by introducing the asymptotic solutions of each structure by the mode-expansion method and direct simulation of the time evolution equation.
This paper investigates the existence of traveling fronts and their propagation speeds for the two component higher order autocatalytic reaction-diffusion systems with any diffusion coefficients. Our elementary analysis of the vector fields in the phase space gives the estimate of the minimal propagation speeds in terms of the order of autocatalysis and the diffusion coefficients.
Global behavior of B models is discussed. When the source term for new B cells equals zero, the system has a conservation quantity. It implies the structurally unstability. It suggests that lack of the source of new B cells may unstabilize the immune system. When the B model incorporates autoimmunity, it loses symmetry. The asymmetry suggests the transition from a tolerant state to autoimmune state is more likely than the inverse transition. Effect of dose of antigen is also considered.
We examine the effect of finite size of population on the distribution of family names. As the result we observe that the power-law behavior of size-frequency distribution in Reed--Hughes () model collapses to show the convex shape on the logarithmic graph. We can approximately calculate the average distribution of size-frequency distribution of family names obtained by the similar method for Ewens sampling formula.