Effects of dispersal on the stability of a prey-predator system with functional response

Using Liapunov's direct method, effects of dispersal on the linear and nonlinear stability of the equilibrium state for a prey-predator system with functional response are investigated. It is noted that the functional response has a destabilizing effect. It is shown that an otherwise linearly or nonlinearly stable equilibrium state of the system remains so with dispersal as well, even with functional response. It is further established that if the equilibrium state is linearly stable a subregion of the positive quadrant can be found in the phase plane where it is nonlinearly stable with or without dispersal.     

Effects of dispersal in a tri-trophic metapopulation model

The aim of this paper is to understand how dispersal in a patchy environment influences the stability properties of tri-trophic metapopulations. Differential equation models for tri-trophic metapopulations are formulated and analysed. The patchy nature of the metapopulations is incorporated through dispersal phases. Two variants are studied: one with a dispersal phase for the top and one with a dispersal phase for the middle level. A complete characterisation of stable and unstable equilibria is given and the possibility of invasion in these food chains is studied. A dispersal phase for the middle level can destabilize the bottom level-middle level interaction, because of the delay that dispersal causes in the reaction to the resource. When the middle level is not efficiently controlled by the top level, the unstable bottom level-middle level pair can destabilize the entire food chain. Dispersal for the top level can destabilize in the same way. A characterisation of the long term behaviour of the models is given. Bistability with a stable three species equilibrium and a stable limit cycle is one of the possibilities.     

Density-dependent dispersal and relative dispersal affect the stability of predator-prey metacommunities

Although density-dependent dispersal and relative dispersal (the difference in dispersal rates between species) have been documented in natural systems, their effects on the stability of metacommunities are poorly understood. Here we investigate the effects of intra- and interspecific density-dependent dispersal on the regional stability in a predator-prey metacommunity model. We show that, when the dynamics of the populations reach equilibrium, the stability of the metacommunity is not affected by density-dependent dispersal. However, the regional stability, measured as the regional variability or the persistence, can be modified by density-dependent dispersal when local populations fluctuate over time. Moreover these effects depend on the relative dispersal of the predator and the prey. Regional stability is modified through changes in spatial synchrony. Interspecific density-dependent dispersal always desynchronizses local dynamics, whereas intraspecific density-dependent dispersal may either synchronize or desynchronize it depending on dispersal rates. Moreover, intra- and interspecific density-dependent dispersal strengthen the top-down control of the prey by the predator at intermediate dispersal rates. As a consequence the regional stability of the metacommunity is increased at intermediate dispersal rates. Our results show that density-dependent dispersal and relative dispersal of species are keys to understanding the response of ecosystems to fragmentation.     

Effects of asymmetric dispersal and environmental gradients on the stability of host–parasitoid systems

We consider host–parasitoid systems spatially distributed on a row of patches connected by dispersal. We analyze the effects of dispersal frequency, dispersal asymmetry, number of patches and environmental gradients on the stability of the host–parasitoid interactions. To take into account dispersal frequency, the hosts and parasitoids are allowed to move from one patch to a neighboring patch a certain number of times within a generation. When this number is high, aggregation methods can be used to simplify the proposed initial model into an aggregated model describing the dynamics of both the total host and parasitoid populations. We show that as the number of patches increases less asymmetric parasitoid dispersal rates are required for stability. We found that the 'CV²>1 rule' is a valid approximation for stability if host growth rate is low, otherwise the general condition of stability we establish should be preferred. Environmental variability along the row of patches is introduced as gradients on host growth rate and parasitoid searching efficiency. We show that stability is more likely when parasitoids move preferentially towards patches where they have high searching efficiency or when hosts go mainly to patches where they have a low growth rate.     

Effects of density-dependent migrations on stability of a two-patch predator-prey model

We consider a predator-prey model in a two-patch environment and assume that migration between patches is faster than prey growth, predator mortality and predator-prey interactions. Prey (resp. predator) migration rates are considered to be predator (resp. prey) density-dependent. Prey leave a patch at a migration rate proportional to the local predator density. Predators leave a patch at a migration rate inversely proportional to local prey population density. Taking advantage of the two different time scales, we use aggregation methods to obtain a reduced (aggregated) model governing the total prey and predator densities. First, we show that for a large class of density-dependent migration rules for predators and prey there exists a unique and stable equilibrium for migration. Second, a numerical bifurcation analysis is presented. We show that bifurcation diagrams obtained from the complete and aggregated models are consistent with each other for reasonable values of the ratio between the two time scales, fast for migration and slow for local demography. Our results show that, under some particular conditions, the density dependence of migrations can generate a limit cycle. Also a co-dim two Bautin bifurcation point is observed in some range of migration parameters and this implies that bistability of an equilibrium and limit cycle is possible.     

Patterns of pollen dispersal in a small population of the Canarian endemic palm (Phoenix canariensis)

The genetic diversity of small populations is greatly influenced by local dispersal patterns and genetic connectivity among populations, with pollen dispersal being the major component of gene flow in many plants species. Patterns of pollen dispersal, mating system parameters and spatial genetic structure were investigated in a small isolated population of the emblematic palm Phoenix canariensis in Gran Canaria island (Canary Islands). All adult palms present in the study population (n=182), as well as 616 seeds collected from 22 female palms, were mapped and genotyped at 8 microsatellite loci. Mating system analysis revealed an average of 5.8 effective pollen donors (N_ep) per female. There was strong variation in correlated paternity rates across maternal progenies (ranging from null to 0.9) that could not be explained by the location and density of local males around focal females. Paternity analysis revealed a mean effective pollen dispersal distance of ∼71 m, with ∼70% of effective pollen originating from a distance of <75 m, and 90% from <200 m. A spatially explicit mating model indicated a leptokurtic pollen dispersal kernel, significant pollen immigration (12%) from external palm groves and a directional pollen dispersal pattern that seems consistent with local altitudinal air movement. No evidence of inbreeding or genetic diversity erosion was found, but spatial genetic structure was detected in the small palm population. Overall, the results suggest substantial pollen dispersal over the studied population, genetic connectivity among different palm groves and some resilience to neutral genetic erosion and subsequently to fragmentation.     

Testing a mechanistic dispersal model against a dispersal experiment with a wind‐dispersed moss

Wind is the main dispersal agent for a wide array of species and for these species the environmental conditions under which diaspores are released can potentially modify the dispersal kernel substantially. Little is known about how bryophytes regulate spore release, but conditions affecting peristome movements and vibration of the seta may be important. We modelled airborne spore dispersal of the bryophyte species Discelium nudum (spore diameter 25 μm), in four different release scenarios, using a Lagrangian stochastic dispersion model and meteorological data. We tested the model predictions against experimental data on colonization success at five distances (5, 10, 30, 50 and 100 m) and eight directions from a translocated point source during seven two‐day periods. The model predictions were generally successful in describing the observed colonization patterns, especially beyond 10 m. In the laboratory we established spore release thresholds; horizontal wind speed sd > 0.25 m s^?1 induced the seta to vibrate and in relative humidity < 75% the peristome was open. Our dispersal model predicts that the proportion of spores dispersing beyond 100 m is almost twice as large if the spores are released under turbulent conditions than under more stable conditions. However, including release thresholds improved the fit of the model to the colonization data only minimally, with roughly the same amount of variation explained by the most constrained scenario (assuming both vibration of the seta and an open peristome) and the scenario assuming random release. Model predictions under realised experimental conditions suggest that we had a low statistical power to rank the release scenarios due to the lack of measurements of the absolute rate of spore release. Our results hint at the importance of release conditions, but also highlight the challenges in dispersal experiments intended for validating mechanistic dispersal models.     

Effect of predator density dependent dispersal of prey on stability of a predator-prey system

This work presents a predator-prey Lotka-Volterra model in a two patch environment. The model is a set of four ordinary differential equations that govern the prey and predator population densities on each patch. Predators disperse with constant migration rates, while prey dispersal is predator density-dependent. When the predator density is large, the dispersal of prey is more likely to occur. We assume that prey and predator dispersal is faster than the local predator-prey interaction on each patch. Thus, we take advantage of two time scales in order to reduce the complete model to a system of two equations governing the total prey and predator densities. The stability analysis of the aggregated model shows that a unique strictly positive equilibrium exists. This equilibrium may be stable or unstable. A Hopf bifurcation may occur, leading the equilibrium to be a centre. If the two patches are similar, the predator density dependent dispersal of prey has a stabilizing effect on the predator-prey system.     

Fig wasp dispersal and the stability of a keystone plant resource in Borneo

The stability of interactions in remaining rainforest fragments is an issue of considerable concern for conservation. Figs are a pre-eminent tropical keystone resource because of their importance for wildlife, but are dependent on tiny (1-2 mm) species-specific wasps for pollination. To investigate fig wasp dispersal I trapped insects at various heights (5-75 m) in an isolated fragment (ca. 4500 ha) of Bornean rain forest. Fig wasps constituted the majority of captures above the canopy (pollinators 47%, non-pollinators 5%). However, genera were not evenly represented. There were 50% more species of monoecious fig pollinator than there were host species in the fragment, indicating some must have arrived from forests with different assemblages of figs at least 30 km away. Dioecious fig pollinators were poorly represented suggesting more limited dispersal, which could account for higher endemism and vulnerability to catastrophic disturbance in these figs. Diurnal activity and flight height also varied among genera. Most non-pollinating fig wasps were very rare.     

Evolutionary stability of ideal free nonlocal dispersal

We study the evolutionary stability of nonlocal dispersal strategies that can produce ideal free population distributions, that is, distributions where all individuals have equal fitness and there is no net movement of individuals at equilibrium. We find that the property of producing ideal free distributions is necessary and often sufficient for evolutionary stability. Our results extend those already developed for discrete diffusion models on finite patch networks to the case of nonlocal dispersal models based on integrodifferential equations. The analysis is based on the use of comparison methods and the construction of sub- and supersolutions.     

Evolutionary stability of ideal free nonlocal dispersal

We study the evolutionary stability of nonlocal dispersal strategies that can produce ideal free population distributions, that is, distributions where all individuals have equal fitness and there is no net movement of individuals at equilibrium. We find that the property of producing ideal free distributions is necessary and often sufficient for evolutionary stability. Our results extend those already developed for discrete diffusion models on finite patch networks to the case of nonlocal dispersal models based on integrodifferential equations. The analysis is based on the use of comparison methods and the construction of sub- and supersolutions.     

Existence, multiplicity and stability of endemic states for an age-structured S-I epidemic model

We study an S-I type epidemic model in an age-structured population, with mortality due to the disease. A threshold quantity is found that controls the stability of the disease-free equilibrium and guarantees the existence of an endemic equilibrium. We obtain conditions on the age-dependence of the susceptibility to infection that imply the uniqueness of the endemic equilibrium. An example with two endemic equilibria is shown. Finally, we analyse numerically how the stability of the endemic equilibrium is affected by the extra-mortality and by the possible periodicities induced by the demographic age-structure.     

Stability in a metapopulation model with density-dependent dispersal

A spatially explicit metapopulation model with positive density-dependent migration is analysed. We obtained conditions under which a previously stable system can be driven to instability caused by a density-dependent migration mechanism. The stability boundary depends on the rate of increase of the number of migrants on each site at local equilibrium, on the intrinsic rate of increase at local level, on the number of patches, and on topological aspects regarding the connectivity between patches. A concrete example is presented illustrating the dynamics on the dispersal-induced unstable regime.     

On a population pathogen model incorporating species dispersal with temporal variation in dispersal rate

In the present paper, we consider a mathematical model of ecosystem population interaction where the population suffers from a susceptible–infectious–susceptible disease. Dispersal of both the susceptible and the infective is incorporated using reaction–diffusion equations. We first study the stability criteria of the basic (non-spatial) model around the disease-free and the infected steady states. We find that the loss rate of the infective species controls disease prevalence. Also without predation pressure, the disease will continue to exist among the population. Then we analyze the spatial model with species dispersal in constant as well as in time-varying form. It is observed that though constant dispersal is unable to generate diffusion-driven instability, dispersal with sinusoidal variation in dispersion rate can generate diffusive instability when the wave number of the perturbation lies within a given range. Numerical simulations are performed to illustrate analytical studies.     

Effects of seed dispersal on Juniperus communis recruitment on a Mediterranean mountain

Abstract. The recruitment of the relict shrub Juniperus communis on a mountain in SE Spain was studied during the period 1994–1998. The main objective was to determine both the quantitative and qualitative effects of bird dispersal on seedling establishment. Seed removal by birds, seed rain, post‐dispersal seed predation, germination, and seedling emergence and survival were analysed in different microhabitats. Birds removed 53 ‐ 89% of the seeds produced by plants. Seed rain was spatially irregular as most seeds accumulated near stones used by birds as perches and below mother plants while a few seeds were dropped in wet meadows and open ground areas. Post‐dispersal seed predation by rodents affected < 10% of dispersed seeds but varied significantly among microhabitats. Only 3.6 ‐ 5.5% of dispersed seeds appeared viable, as many seeds had aborted or showed wasp damage. Seeds germinated in the second and third springs after sowing, reaching a germination percentage of 36%. Seedling emergence was concentrated in wet meadows. Seedling mortality was high (75–80%), but significantly lower in wet meadows, the only microhabitat where seedlings could escape from summer drought, the main mortality cause. Seed abortion, germination and seedling mortality proved to be the main regeneration constraints of J. communis on Mediterranean mountains. Birds exerted a strong demographic effect, although their qualitative effect was limited by abiotic factors which caused the pattern of seed rain to differ from the final pattern of recruitment between microhabitats.     

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.     

Seed dispersal and germination behavior of three threatened endemic labiates of Cyprus

The present study examined seed dispersal and germination in three of the most threatened endemic labiates of Cyprus: Origanum cordifolium , Phlomis brevibracteata and Phlomis cypria ssp. occidentalis . Some common traits in these taxa can be correlated with their overall survival strategy. Seeds mature in mid to late summer, but most seeds remain on the mother plants until the beginning of the rainy season. The opening of the calyces containing the seeds seems to be caused by absorption of moisture. Water is also the most important dispersal factor because the seeds are dispersed by rain. Seed germination occurs at relatively low temperatures that prevail in the field at the beginning of the rainy season. This behavior provides the plants with ecological advantages because their seeds are exposed on the soil at the most appropriate period (mid to late autumn) for germination and seedling survival. The present study contributes substantially to in situ and ex situ conservation of these threatened plants.