A note on population growth under random dispersal

An approximation method using a sine function is used to solve the second degree growth equation for the case in which an organism may simultaneously become dispersed throughout a uniform region. The resulting expression for a special case is compared with the expression obtained by R. Barakat (1959,Bull. Math. Biophysics,21, 141–51), giving the first two terms, by an iterative, procedure. The agreement is satisfactory.     

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.     

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 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     

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     

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.     

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.     

The evolution of dispersal in random environment

In this paper we introduce a stochastic model for a population living and migrating between s sites without distinction in the states between residents and immigrants. The evolutionary stable strategies (ESS) is characterized by the maximization of a stochastic growth rate. We obtain that the expectation of reproductive values, normalized by some random quantity, are constant on all sites and that the expectation of the normalized vector population structure is proportional to eigenvector of the dispersion matrix associated to eigenvalue one, which are, in some way, analogous to the results obtained in the deterministic case.     

General population growth models with Allee effects in a random environment

Allee effects on population growth are quite common in nature, usually studied through deterministic models with a specific growth rate function.In order to seek the qualitative behaviour of populations induced by such effects, one should avoid model-specific behaviours. So, we use as a basis a general deterministic model, i.e. a model with a general growth rate function, to which we add the effect on the growth rate of the random fluctuations in environmental conditions. The resulting model is the general stochastic differential equation (SDE) model that we propose here.We consider two possible cases, weak Allee effects and strong Allee effects, which lead to different qualitative behaviours of the model.We will study the model properties for both cases in terms of existence and uniqueness of the solution, extinction and stationary behaviour of the population. The two cases will be compared with each other and with the general density-dependent SDE model without Allee effects.We then consider as an example the particular case of the classic logistic model and an Allee effect version of it.     

Impact of dispersal status on estimates of local population growth rates in a Temminck's stint Calidris temminckii population

Due to different costs and benefits associated with dispersal and philopatry, life history traits of immigrants and philopatric individuals may differ. Despite of the apparent effects, dispersal status is only rarely considered in analyses of population dynamics. We analysed whether dispersal status explains variation in life history traits of an endangered Temminck's stint Calidris temminckii population breeding at the Baltic Sea. We also estimated the impact of immigration and dispersal status on the population growth rate (λ) with a population matrix model, in which immigrants and philopatric individuals are separated to their own stages. We found that philopatric individuals had a higher apparent survival than immigrants in both sexes. In reproductive parameters, variation due to dispersal status was not clear. Nests incubated by philopatric individuals survived better than those of immigrants, but this did not translate in hatchling production per breeding attempt. Models described a sink population in which the inclusion of both immigration rate to the population and the dispersal status of individuals into the model increased estimates of λ. When the better success of philopatric individuals was considered, the population growth appeared more stable (λ=0.972). If this was not taken into account, λ implied a strong decline (λ=0.911). The results support the hypothesis that immigrants exhibit lower components of lifetime reproductive success and therefore contribute less to population growth and the gene pool than local recruits. While we cannot distinguish whether this difference reflects higher mortality or permanent emigration, the latter explanation seems more plausible. Our results highlight the importance of considering immigration and dispersal status in population modelling. In the case of the endangered study population, the results implied that management options directed to improve local recruitment would be a profitable option.     

Symbiotic feather mites synchronize dispersal and population growth with host sociality and migratory disposition

Some symbiotic taxa may have evolved to track changes in the level and quality of food resources provided by the host to increase reproduction and dispersal. As a consequence, some ectosymbionts synchronize their reproduction and activity with particular stages of their host's living cycle. In this article we examined temporal patterns of variation in prevalence and abundance of feather mites living on pre‐migratory barn swallows Hirundo rustica. Feather mites in the lineages Pterolichoidea and Analgoidea are the most common arthropod ectosymbionts living at the expenses of feather oil. We investigated whether the seasonal variations in levels of several measures of physiological condition associated with host migration were related to changes in prevalence and abundance of mites. The results suggest that the variation in prevalence of feather mites, and thus probably the mode of acquisition and dispersal of these symbionts, is linked to an increase in host sociality before migration. Physiological dynamics of hosts after the breeding season point at two clearly identifiable periods: a post‐breeding period when physiological condition remains stationary or decreases, and a pre‐migratory period characterized by a rapid increase in several measures of physiological condition. Mite population dynamics were synchronized with migratory disposition during the period of highest host gregariousness. These synchronized processes occurred in both study years, although dynamics of migratory disposition and mite prevalence and abundance differ somewhat between years for adult and juvenile hosts. Mite population increase before host migration may be a response to a higher quantity of food provided by the host, namely oil from the urpoygial gland which is stimulated by hormones. Therefore, mites might have evolved to adjust their reproduction to the time when they have more chance of dispersal through horizontal transmission. In addition, body mass of juvenile and adult hosts were positively related with mite abundance in both years after allowing for several influencing factors. Body mass variation may reflect adequately fitness of host or their current physiological state, for instance, differences in the secretion of lipids on feathers or a more adequate microclimate to these symbionts.     

Density-dependent dispersal and spatial population dynamics

The synchronization of the dynamics of spatially subdivided populations is of both fundamental and applied interest in population biology. Based on theoretical studies, dispersal movements have been inferred to be one of the most general causes of population synchrony, yet no empirical study has mapped distance-dependent estimates of movement rates on the actual pattern of synchrony in species that are known to exhibit population synchrony. Northern vole and lemming species are particularly well-known for their spatially synchronized population dynamics. Here, we use results from an experimental study to demonstrate that tundra vole dispersal movements did not act to synchronize population dynamics in fragmented habitats. In contrast to the constant dispersal rate assumed in earlier theoretical studies, the tundra vole, and many other species, exhibit negative density-dependent dispersal. Simulations of a simple mathematical model, parametrized on the basis of our experimental data, verify the empirical results, namely that the observed negative density-dependent dispersal did not have a significant synchronizing effect.     

Reduction in population growth under different contraceptive policies

A number of equations for the various population control policies are worked out for a desired reduction in the rate of growth. At the ages of 25 and 30 respectively, 61 and 97% of contraceptive users are necessary to reduce the present rate of growth of 0.026 to 0.010. While at the age of 25 about 69 and 76% contraceptive users are required for the same reduction in the rate of growth, assuming that 25 and 50% would discontinue the use of contraceptives at the age of 35. The birth and death rates in the study area (Varanasi Rural) have remained almost constant for several years, justifying the assumption of a stable population. This study emphasises the need for the use of contraceptive devices at two or more age levels.     

Contributions of long‐distance dispersal to population growth in colonising Pinus ponderosa populations

Long‐distance dispersal is an integral part of plant species migration and population development. We aged and genotyped 1125 individuals in four disjunct populations of Pinus ponderosa that were initially established by long‐distance dispersal in the 16th and 17th centuries. Parentage analysis was used to determine if individuals were the product of local reproductive events (two parents present), long‐distance pollen dispersal (one parent present) or long‐distance seed dispersal (no parents present). All individuals established in the first century at each site were the result of long‐distance dispersal. Individuals reproduced at younger ages with increasing age of the overall population. These results suggest Allee effects, where populations were initially unable to expand on their own, and were dependent on long‐distance dispersal to overcome a minimum‐size threshold. Our results demonstrate that long‐distance dispersal was not only necessary for initial colonisation but also to sustain subsequent population growth during early phases of expansion.     

Phoretic dispersal influences parasite population genetic structure

Dispersal is a fundamental component of the life history of most species. Dispersal influences fitness, population dynamics, gene flow, genetic drift and population genetic structure. Even small differences in dispersal can alter ecological interactions and trigger an evolutionary cascade. Linking such ecological processes with evolutionary patterns is difficult, but can be carried out in the proper comparative context. Here, we investigate how differences in phoretic dispersal influence the population genetic structure of two different parasites of the same host species. We focus on two species of host‐specific feather lice (Phthiraptera: Ischnocera) that co‐occur on feral rock pigeons (Columba livia). Although these lice are ecologically very similar, “wing lice” (Columbicola columbae) disperse phoretically by “hitchhiking” on pigeon flies (Diptera: Hippoboscidae), while “body lice” (Campanulotes compar) do not. Differences in the phoretic dispersal of these species are thought to underlie observed differences in host specificity, as well as the degree of host–parasite cospeciation. These ecological and macroevolutionary patterns suggest that body lice should exhibit more genetic differentiation than wing lice. We tested this prediction among lice on individual birds and among lice on birds from three pigeon flocks. We found higher levels of genetic differentiation in body lice compared to wing lice at two spatial scales. Our results indicate that differences in phoretic dispersal can explain microevolutionary differences in population genetic structure and are consistent with macroevolutionary differences in the degree of host–parasite cospeciation.     

Effects of dispersal on local population increase

Work to date on biological invasions and the spread of biological control agents has been focused on the explicitly spatial aspects, such as rate of spread and shape of the wave front. There has been relatively little attention paid to the influence of dispersal on the rate of increase of local populations. We use a simple general model for logistic local growth and one-dimensional diffusive dispersal to show that dispersal can act as a substantial drain on local populations. Local increase at a site of introduction is always slower than would be expected in the absence of dispersal, while the rate of increase of other populations is initially enhanced, then reduced by dispersal. This may have an important effect when estimating population parameters for invading organisms or biological control agents during the initial stages of their spread, and helps explain the "latent period" typically observed in biological invasions.     

A stationary distribution for the growth of a population subject to random catastrophes

The problem of the existence of a stationary distribution and the convergence towards it in a certain semistochastic model for the growth of a population is considered. It is assumed that the population grows according to a deterministic equation, but at certain times there are catastrophes, which lead to a decrease in the population level. The hazard function for the occurrence of catastrophes is a function of the population level only. The size of these jumps have a distribution that depends on the population size immediately before the catastrophe. A constructive method for finding the stationary distribution is given.     

Two-patch population models with adaptive dispersal: the effects of varying dispersal speeds

The population-dispersal dynamics for predator–prey interactions and two competing species in a two patch environment are studied. It is assumed that both species (i.e., either predators and their prey, or the two competing species) are mobile and their dispersal between patches is directed to the higher fitness patch. It is proved that such dispersal, irrespectively of its speed, cannot destabilize a locally stable predator–prey population equilibrium that corresponds to no movement at all. In the case of two competing species, dispersal can destabilize population equilibrium. Conditions are given when this cannot happen, including the case of identical patches.