Logistic population growth under random dispersal

Various diffusion processes employed for modelling logistic growth are briefly summarized. A discrete-time, discrete-state space stochastic process for population growth is proposed and analyzed with either Bose-Einstein or Maxwell-Boltzmann statistics for the distribution of offspring in available sites in a restricted region. A diffusion approximation is constructed, which differs from those previously employed. The logistic law is a natural deterministic analog of the diffusion process.     

Population growth under the influence of random dispersal

A population is considered which grows according to the logistic equation while spreading out at random. An approximate method is used to obtain transient and steady-state values for various simple boundary conditions such as that of a population started in an infinite one- or two-dimensional region with or without reflecting or absorbing barriers. An approximate steady-state solution is given for the one-dimensional case of two neighboring regions having different growth rates, mobilities, and degrees of attractiveness.     

A note on growth processes in random environment

The purpose of this note is to discuss certain features of population growth models carlier proposed and to construct an alternative diffusion model for regulated growth in random environment. This model is shown to be the analogue of the Malthusian one, although it is a generalization of the latter due to the presence of regulation.     

A note on the transient stage of the random dispersal of logistic populations

The transient stage of the random dispersal of logistic populations is investigated, using a Sturm-Liouville series leading to an infinite system of non-linear integral equations. These equations are then solved via a successive approximation scheme. R. A. Fisher's (steady-state) velocity of advance paradox is discussed. An illustrative example is worked to the second order of approximation. Contribution No. 1020 of the Woods Hole Oceanographic Institution.     

A generalized diffusion model for growth and dispersal in a population

A reaction-diffusion model is presented in which spatial structure is maintained by means of a diffusive mechanism more general than classical Fickian diffusion. This generalized diffusion takes into account the diffusive gradient (or gradient energy) necessary to maintain a pattern even in a single diffusing species. The approach is based on a Landau-Ginzburg free energy model. A problem involving simple logistic kinetics is fully analyzed, and a nonlinear stability analysis based on a multi-scale perturbation method shows bifurcation to non-uniform states.Part of this work was done while at the Mathematical Institute, Oxford University as a Senior Visiting Fellow supported by the Science Research Council of Great Britain under grant GR/B31378     

A diffusion model for population growth in random environment

The growth of a population in a randomly varying environment is modeled by replacing the Malthusian growth rate with a delta-correlated normal process. The population size is then shown to be a random process, lognormally distributed, obeying a diffusion equation of the Fokker-Planck type. The first passage time p.d.f. through any arbitrarily assigned value and the probability of absorption are derived. The asymptotic behavior of the population size is investigated.     

Environmentally induced dispersal under heterogeneous logistic growth

We consider a single-species model which is composed of several habitats connected by linear migration rates and having logistic growth. A spatially varying, temporally constant environment is introduced by the non-homogeneity of its carrying capacity. Under this condition any type of purely diffusive behavior, characterized in our model by symmetric migration rates, produces an unbalanced population distribution, i.e. some locations receive more individuals than can be supported by the environmental carrying capacity, while others receive less. Using an evolutionarily stable strategy (ESS) approach we show that an asymmetric migration mechanism, induced by the heterogeneous carrying capacity of the environment, will be selected. This strategy balances the inflow and outflow of individuals in each habitat (balanced dispersal), as well as 'balancing' the spatial distribution relative to variation in carrying capacity (the Ideal Free Distribution from habitat selection theory). We show that several quantities are maximized or minimized by the evolutionarily stable dispersal strategy.     

A note on distributions of times to coalescence,under time-dependent population size

Expressions for marginal distributions of times in the time-varying coalescence process are derived. The proposed method allows also for computation of joint probability distribution for pairs, triples, etc. of coalescence times. The expressions derived are useful for (1) extending several statistics from time constant to time-varying case, (2) increasing efficiency and accuracy of simulations in time-varying evolution, and (3) debugging coalescence simulation software.     

A note on type II error under random effects misspecification in generalized linear mixed models

Litière, Alonso, and Molenberghs (2007, Biometrics, 63, 1038-1044) presented the results of simulation studies that they claimed showed that misspecification of the shape of the random effects distribution can produce marked increases in Type II error (decreases in power) of tests based on fits of generalized linear mixed models. However, the article contains a logical fallacy that invalidates this claim. We present logically correct simulation studies that demonstrate little increase in Type II error, consistent with the earlier work that shows little effect due to misspecification.     

Impact of dispersal on population growth: the role of inter-patch distance

Dispersal can play a key role in the dynamics of patchy populations through patch colonization, and generally this leads to distance-dependent colonization. Less recognised are the roles of dispersal and inter-patch distance on the growth of a population after colonization. We use a laboratory mite model system in which both juveniles and adults can disperse to explore the impact of dispersal, and particularly inter-patch distance, on population dynamics. We examine the dynamics of patches after colonization by manipulating the presence of a dispersal corridor to a source patch at two inter-patch distances. Consistent with many field studies, the results show colonization was slower in more distant patches. Following colonization, the effect of the dispersal corridor on dynamics was dependent on inter-patch distance. In patches near the source, the number of adults tended to increase at a faster rate, and juveniles at a slower rate when connected with a dispersal corridor. In contrast, adult numbers grew slower and juveniles tended to grow faster when connected with a corridor in more distant patches. In the long-term, equilibrium adult numbers were lower in patches connected to the source patch at both distances. These results are likely to be driven by the effects of inter-patch distance on dispersal mortality, and the effects of dispersal on patch abundance and within-patch competition. These results confirm that distance is important for patch colonization and also show that distance can affect population density after colonization. The effects of dispersal and distance on local dynamics could be important in the dynamics of patchy populations in increasingly fragmented landscapes.     

A note on craniofacial sutural growth

The paper presents and discusses some features of sutural structures. Fibre orientation in sutures seems to be quite variable and associated with minute local growth phenomena. Trabecular patterns in bones, reflecting growth, appear to support the idea of the outside determination of growth. The bevelled and interdigitated structures seen in many sutures may be interpreted as an expedient solution to the problem of fast growth.     

A note on stability of discrete population models

P. Cull (1981) and G. Rosencranz (1983) studied a discrete population model described by the first-order difference equation xt+1 = g(xt) and obtained an important result on the global stability of the equilibrium point means when g(x) has only one extreme point (a maximum) in (0, means). Motivated by work of M. Kot and W. M. Schaffer (1984), a more general case is considered in which g(x) can have more then one maximum point in (0, means), and results on global stability are obtained. These results are applied to develop tests for global stability of the equilibrium point that imply other results in the literature on global stability.     

A note on persistence about structured population models

In this paper, we report some results on persistence in two structured population models: a chronic- age-structured epidemic model and an age-duration-structured epidemic model. Regarding these models, we observe that the system is uniformly strongly persistent, which means, roughly speaking, that the proportion of infected subpopulation is bounded away from 0 and the bound does not depend on the initial data after a sufficient long time, if the basic reproduction ratio is larger than one. We derive this by adopting Thieme's technique, which requires some conditions about positivity and compactness. Although the compactness condition is rather difficult to show in general infinite-dimensional function spaces, we can apply Fréchet-Kolmogorov L(1)-compactness criteria to our models. The two examples that we study illuminate a useful method to show persistence in structured population models.     

A note on persistence about structured population models

In this paper, we report some results on persistence in two structured population models: a chronic- age-structured epidemic model and an age-duration-structured epidemic model. Regarding these models, we observe that the system is uniformly strongly persistent, which means, roughly speaking, that the proportion of infected subpopulation is bounded away from 0 and the bound does not depend on the initial data after a sufficient long time, if the basic reproduction ratio is larger than one. We derive this by adopting Thieme's technique, which requires some conditions about positivity and compactness. Although the compactness condition is rather difficult to show in general infinite-dimensional function spaces, we can apply Fréchet–Kolmogorov L ¹-compactness criteria to our models. The two examples that we study illuminate a useful method to show persistence in structured population models.     

A note on the problem of cycles in random nets

The problem of cycles in random neural nets is considered. An approximation is derived giving the expected numberE(k, a, n) of cycles of lengthk of which a neuron in a random net will be a member. The approximation is further simplifed for the casen> k. In this caseE(k, a, n)=a ^k/n , where (a) is the axone density of the net.     

On a conjecture concerning population growth in random environment

Discrete stochastic models are constructed and their limit diffusion processes are derived to shed light on a controversial conjecture regarding the effects of environmental variance on the asymptotic behavior of a population subject to logistic growth in random environment.Work supported in part by the National Group for Mathematical Information Sciences (GNIM) of the National Council for Research