Symmetric Dispersal Research Articles

Bounding Seed Loss from Isolated Habitat Patches

Dispersal of propagules (seeds, spores) from a geographically isolated habitat into an uninhabitable matrix can play a decisive role in driving population dynamics. ODE and integrodifference models of these dynamics commonly feature a “dispersal success” parameter representing the average proportion of dispersing propagules that remain in viable habitat. While dispersal success can be estimated by empirical measurements or by integration of dispersal kernels, one may lack resources for fieldwork or details on dispersal kernels for numerical computation. Here we derive simple upper bounds on the proportion of propagule loss—the complement of dispersal success—that require only habitat area, habitat perimeter, and the mean dispersal distance of a propagule. Using vector calculus in a probabilistic framework, we rigorously prove bounds for the cases of both symmetric and asymmetric dispersal. We compare the bounds to simulations of integral models for the population of Asclepias syriaca (common milkweed) at McKnight Prairie—a 14 hectare reserve surrounded by agricultural fields in Goodhue County, Minnesota—and identify conditions under which the bounds closely estimate propagule loss.

The Effects of Varying Predator Dispersal Strength on Prey-Predator Dynamics with Refuge Process

The combined effects of (symmetric or asymmetric) dispersal and refuge mechanisms can have a significant impact on prey-predator dynamics. However, there remains a knowledge gap in concerning the incorporation of asymmetrical dispersal in the presence of prey refuges. Therefore, this paper aims to examine the influence of varying levels of asymmetrical (i.e., predator) dispersal on the interactions between prey and predators, as well as the role of prey refuges in facilitating species coexistence. The investigation begins by introducing an ordinary differential equation (ODE) model for the prey-predator system, which is subsequently extended to a partial differential equation (PDE) model. By conducting one-parameter bifurcation analysis on both models, the presence of transcritical and Hopf bifurcation points is established. Furthermore, the research delves into the spatio-temporal dynamics of the PDE model, capturing the intricate interactions between a specialized prey species and its predator. The focus is on examining the effects of different strengths of predator dispersal on the dynamics of the prey-predator system. The aim is to gain a comprehensive understanding of how predator dispersal influences the stability and persistence of the system, and to investigate the ecological implications of these dynamics in terms of prey-predator coexistence. Hence, the main findings of the research suggest that the increased levels of predator dispersal led to a wider range of prey refuges, supporting species coexistence. In conclusion, this study emphasises the critical importance of predator and prey dispersal dynamics in determining the key mechanisms that can promote species coexistence

The effect of dispersal on asymptotic total population size in discrete- and continuous-time two-patch models

Many populations occupy spatially fragmented landscapes. How dispersal affects the asymptotic total population size is a key question for conservation management and the design of ecological corridors. Here, we provide a comprehensive overview of two-patch models with symmetric dispersal and two standard density-dependent population growth functions, one in discrete and one in continuous time. A complete analysis of the discrete-time model reveals four response scenarios of the asymptotic total population size to increasing dispersal rate: (1) monotonically beneficial, (2) unimodally beneficial, (3) beneficial turning detrimental, and (4) monotonically detrimental. The same response scenarios exist for the continuous-time model, and we show that the parameter conditions are analogous between the discrete- and continuous-time setting. A detailed biological interpretation offers insight into the mechanisms underlying the response scenarios that thus improve our general understanding how potential conservation efforts affect population size.

Mathematical modeling of the population dynamics of a distinct interactions type system with local dispersal

Distinct biotic interactions in multi-species communities are a ubiquitous force in the natural ecosystem, and this force is an essential determinant of community stability and species coexistence outcomes. We conduct numerical simulations and bifurcation analysis of partial differential equations to gain better understanding and ecological insights into how predation (a), predator handling time (h), and local dispersal affect multi-species community dynamics. This system consists of resource-mutualist-exploiter-competitor interactions and local dispersal. From the inspection of our numerical simulations and co-dimension one bifurcation analysis findings, we discover several critical values that correspond to transcritical bifurcation, subcritical and supercritical Hopf bifurcations. This occurs as we vary the bifurcation parameters a and h in this complex ecological system under symmetric and asymmetric dispersal scenarios. Furthermore, the interplay between these local bifurcation points results in an exciting co-dimension two bifurcations, i.e., Bogdanov-Takens and cusp bifurcation points, respectively, which act as the synchronization points in this complex ecological system. From an ecological viewpoint, we find that (i) the effect of the no-dispersal scenario supports the maintenance of species biodiversity when the predation strength is moderate; (ii) symmetric dispersal induces both subcritical and supercritical Hopf bifurcation and support species diversity for moderate predation strength; and (iii) asymmetric dispersal promotes species diversity as it simplifies the bifurcation changes in dynamics by eliminating the subcritical bifurcations that trigger uncertainty, and this dispersal mechanism mediates species coexistence outcomes. Fundamentally, stable limit cycles have been reported as predator handling time varies in some ecological models; however, we observed in our bifurcation analysis the emergence of the unstable limit cycle as predator handling time changes. We discover that intense predator handling time destabilizes this complex ecological community. In general, our results demonstrate the influential roles of predation, predator handling time, and local dispersal in determining this system’s coexistence dynamics. This knowledge provides a better understanding of species conservation and biological control management.

Volumetric Mapping of Methane Concentrations at the Bush Hill Hydrocarbon Seep, Gulf of Mexico

The role of methane as a green-house gas is widely recognized and has sparked considerable efforts to quantify the contribution from natural methane sources including submarine seeps. A variety of techniques and approaches have been directed at quantifying methane fluxes from seeps from just below the sediment water interface all the way to the ocean atmosphere interface. However, there have been no systematic efforts to characterize the amount and distribution of dissolved methane around seeps. This is critical to understanding the fate of methane released from seeps and its role in the submarine environment. Here we summarize the findings of two field studies of the Bush Hill mud volcano (540 m water depth) located in the Gulf of Mexico. The studies were carried out using buoyancy driven gliders equipped with methane sensors for near real time in situ detection. One glider was equipped with an Acoustic Doppler Current Profiler (ADCP) for simultaneous measurement of currents and methane concentrations. Elevated methane concentrations in the water column were measured as far away as 2 km from the seep source and to a height of about 100 m above the seep. Maximum observed concentrations were ∼400 nM near the seep source and decreased away steadily in all directions from the source. Weak and variable currents result in nearly radially symmetric dispersal of methane from the source. The persistent presence of significant methane concentrations in the water column points to a persistent methane seepage at the seafloor, that has implications for helping stabilize exposed methane hydrates. Elevated methane concentrations in the water column, at considerable distances away from seeps potentially support a much larger methane-promoted biological system than is widely appreciated.

Dispersal asymmetry in a two-patch system with source–sink populations

This paper analyzes source–sink systems with asymmetric dispersal between two patches. Complete analysis on the models demonstrates a mechanism by which the dispersal asymmetry can lead to either an increased total size of the species population in two patches, a decreased total size with persistence in the patches, or even extinction in both patches. For a large growth rate of the species in the source and a fixed dispersal intensity, (i) if the asymmetry is small, the population would persist in both patches and reach a density higher than that without dispersal, in which the population approaches its maximal density at an appropriate asymmetry; (ii) if the asymmetry is intermediate, the population persists in both patches but reaches a density less than that without dispersal; (iii) if the asymmetry is large, the population goes to extinction in both patches; (iv) asymmetric dispersal is more favorable than symmetric dispersal under certain conditions. For a fixed asymmetry, similar phenomena occur when the dispersal intensity varies, while a thorough analysis is given for the low growth rate of the species in the source. Implications for populations in heterogeneous landscapes are discussed, and numerical simulations confirm and extend our results.

A consumer–resource system with source–sink populations and asymmetric dispersal

In this paper, we consider a two-patch system with source–sink populations and asymmetric dispersal, which includes exploitable resources and extends a recent model describing experiments. Applying dynamical systems theory, we reveal uniform persistence of the system and exhibit existence of stable positive equilibria. Based on rigorous analysis, we demonstrate that dispersal can lead to survival of species in both patches, and asymmetric dispersal can make the species reach higher density than that with symmetric dispersal or with no dispersal. A new prediction of this paper is that in source–sink populations, the species with asymmetric dispersal can approach a density higher than that in the corresponding homogeneous resource-distributions with or without dispersal, which extends previous theory. It is shown that small asymmetry to the sink patch can increase total population abundance, while extremely large asymmetry would result in extinction of species. Our findings are consistent with experimental results and provide new predictions. Numerical computations confirm and extend the findings.

Accelerating propagation in a recursive system arising from seasonal population models with nonlocal dispersal

The system of interest is defined by iterations of the Poincaré map of a time periodic population model with symmetric nonlocal dispersal and periodic time delay. Under the KPP type setting, we show that the system admits the spreading speed c⁎∈(0,+∞] that coincides with the minimal speed of traveling waves, where c⁎<+∞ if the nonlocal dispersal kernel J is exponentially bounded and c⁎=+∞ if J is exponentially unbounded. Further, the latter case gives rises to an accelerating propagation that the level set of solutions with compactly supported initial values locate in between two exponential functions in time.

Dynamics of a population in two patches with dispersal

A two-dimensional discrete system of a species in two patches proposed by Newman et al. is studied. It is shown that the unique interior steady state is globally asymptotically stable if the active population has a Beverton–Holt type growth rate. If the population is also subject to Allee effects, then the system has two interior steady states whenever the density-independent growth rate is large. In addition, the model has period-two solutions if the symmetric dispersal exceeds a critical threshold. For small dispersal, populations may either go extinct or eventually stabilize. However, populations are oscillating over time if dispersal is beyond the critical value and the initial populations are large.

Dynamical effects of nonlocal interactions in discrete-time growth-dispersal models with logistic-type nonlinearities

The paper is devoted to the study of discrete time and continuous space models with nonlocal resource competition and periodic boundary conditions. We consider generalizations of logistic and Ricker's equations as intraspecific resource competition models with symmetric nonlocal dispersal and interaction terms. Both interaction and dispersal are modeled using convolution integrals, each of which has a parameter describing the range of nonlocality. It is shown that the spatially homogeneous equilibrium of these models becomes unstable for some kernel functions and parameter values by performing a linear stability analysis. To be able to further analyze the behavior of solutions to the models near the stability boundary, weakly nonlinear analysis, a well-known method for continuous time systems, is employed. We obtain Stuart–Landau type equations and give their parameters in terms of Fourier transforms of the kernels. This analysis allows us to study the change in amplitudes of the solutions with respect to ranges of nonlocalities of two symmetric kernel functions. Our calculations indicate that supercritical bifurcations occur near stability boundary for uniform kernel functions. We also verify these results numerically for both models.

To connect or not to connect isolated patches.

Empirical evidence suggests that dispersal can have different effects on the time evolution of a spatially structured population. In this study, we explored the impact of the migration rate in a coupled map lattice system. To capture this impact, we assumed symmetric dispersal and simple dynamics in the local populations. However, we allowed heterogeneity between the patches, including both source-source and source-sink systems. Our results show that this simple theoretical setting has the potential to unify the diversity of behaviours of the total population size observed in previous studies. Indeed, we found that the response of the total population size to migration was non-monotone in source-source and many source-sink situations, thereby suggesting that an increase in the dispersal rate could be related to either an increase or a decrease in the total population size. As we will illustrate, this response provides a possible theoretical explanation of some benefits of control strategies involving spatial considerations as no-take zones. Our study also analyses the impact of the migration rate on persistence, spatial coherence, and initial transients. This was motivated by previous theoretical observations in coupled systems that the rate of migration affects these three aspects. Related to persistence, we rigorously extended a previous result from the linear to the non-linear case. This result essentially states that persistence depends on the stability of the origin. On the other hand, we stress that negative effects due to an increase of the spatial coherence could be neutralised by the unimodal response of the total population size. Finally, the study of the initial transients, which is relevant for interpreting experimental results, highlights that the relationship between the rate of migration and the total population size remains the same even in the transient phase.

Dispersal dynamics in food webs.

Studies of food webs suggest that limited nonrandom dispersal can play an important role in structuring food webs. It is not clear, however, whether density-dependent dispersal fits empirical patterns of food webs better than density-independent dispersal. Here, we study a spatially distributed food web, using a series of population-dispersal models that contrast density-independent and density-dependent dispersal in landscapes where sampled sites are either homogeneously or heterogeneously distributed. These models are fitted to empirical data, allowing us to infer mechanisms that are consistent with the data. Our results show that models with density-dependent dispersal fit the α, β, and γ tritrophic richness observed in empirical data best. Our results also show that density-dependent dispersal leads to a critical distance threshold beyond which site similarity (i.e., β tritrophic richness) starts to decrease much faster. Such a threshold can also be detected in the empirical data. In contrast, models with density-independent dispersal do not predict such a threshold. Moreover, preferential dispersal from more centrally located sites to peripheral sites does not provide a better fit to empirical data when compared with symmetric dispersal between sites. Our results suggest that nonrandom dispersal in heterogeneous landscapes is an important driver that shapes local and regional richness (i.e., α and γ tritrophic richness, respectively) as well as the distance-decay relationship (i.e., β tritrophic richness) in food webs.

Impact of Dispersal on the Total Population Size, Constancy and Persistence of Two-patch Spatially-separated Populations

This paper explores by means of extensive numerical simulation how unidirectional and bidirectional symmetric dispersal can affect the mean total population size and the fluctuation range in a two-patch population model described by coupled difference equations. The obtained results show that the response to dispersal varies not only with the type of connection between subpopulations, but also with the intrinsic dynamics in each subpopulation. We find that the mean total population size increases monotonically with unidirectional dispersal from a region with local complicated dynamics to a region with an attracting equilibrium, whereas in the other studied scenarios the response is generically unimodal. Constancy and persistence are considered by relating them to the fluctuation range. Our results show that dispersal is capable of enhancing constancy and persistence only in certain situations. Additionally, we show that multistability affects the behaviour of the mean total population size and its fluctuation range. Hence it can be concluded that, in contrast to other control strategies, only if there is a good knowledge of the local population dynamics, then a modification of the natural dispersal rate between regions might be used as a way to control the size and stability of the population.

Effects of symmetric and asymmetric dispersal on the dynamics of heterogeneous metapopulations: Two-patch systems revisited

Although the effects of dispersal on the dynamics of two-patch metapopulations are well studied, potential interactions between local dynamics and asymmetric dispersal remain unexplored. We examined the dynamics of two Ricker models coupled by symmetric or asymmetric constant-fraction dispersal at different rates. Unlike previous studies, we extensively sampled the r1−r2 space and found that stability of the coupled system was markedly affected by interactions between dispersal (in terms of strength and asymmetry) and local dynamics. When both subpopulations were intrinsically chaotic, increased symmetry in the exchange of individuals had a greater stabilizing impact on the dynamics of the system. When one subpopulation showed considerably more unstable dynamics than the other, higher asymmetry in the exchange of individuals had a stabilizing or destabilizing effect on the dynamics depending on whether the net dispersal bias was from the relatively stable to the relatively unstable subpopulation, or vice versa. The sensitivity of chaotic dynamics to stabilization due to dispersal varied with r-value in the chaotic subpopulation. Under unidirectional or bidirectional symmetric dispersal, when one subpopulation was intrinsically chaotic and the other had stable dynamics, the stabilization of chaotic subpopulations with r~3.3–4.0 occurred at the lowest dispersal rates, followed by chaotic subpopulations with r~2.7–2.95 and, finally, chaotic subpopulations with r~2.95–3.3. The mechanism for this pattern is not known but might be related to the range and number of different attainable population sizes possible in different r-value zones.

Use of multiple habitat types with asymmetric dispersal affects patch occupancy of the damselfly Indolestes peregrinus in a fragmented landscape

To appropriately predict the patch occupancy of animals, it is often essential to consider not only the habitat structure but also shifts in the habitat requirements of animals with changes in life stage. In addition, asymmetric dispersal among different types of habitat patches is likely to accompany use of multiple habitat types due to differences in the ease with which migrants can find the habitats, to changes in the dispersal ability of animals according to their life stage, or to both factors. However, few studies have explicitly elucidated the contribution of these processes to patch connectivity and to predictions of patterns of patch occupancy. In the present study, we evaluated the effects of multi-type habitat use on patch connectivity of the damselfly Indolestes peregrinus. After emergence, adults of this species move from their native ponds to woodlands for hibernation and return to aquatic habitats for oviposition in the next spring. We recorded the occurrence of I. peregrinus at newly created artificial ponds and attempted to explain patch occupancy using a series of Bayesian statistical models, which incorporate (1) local environment only, (2) both local environment and single-type habitat use connectivity, and (3) both local environment and multi-type habitat use connectivity. In addition, we considered two situations in the third model: symmetric or asymmetric dispersal. Comparing the performance of the candidate models revealed that the best model was the third model assuming asymmetric dispersal and it explained 18.8% of the deviance. The result suggests that multi-type habitat use is important for determining patch connectivity of I. peregrinus, and that there is asymmetry in the connectivity from pond to woodland patches and vice versa for the damselfly. Both multi-type habitat use and asymmetric dispersal processes are likely to apply to many other taxa and landscapes.

A Single Species Model with Symmetric Bidirectional Impulsive Diffusion and Dispersal Delay

In the natural ecosystem, impulsive diffusion provides a more natural description for population dynamics. In addition, dispersal processes often involve with time delay. In view of these facts, a single species model with impulsive diffusion and dispersal delay is formulated. By the stroboscopic map of the discrete dynamical system and other analysis methods, the permanence of the system is investigated. Moreover, sufficient conditions on the existence and uniqueness of a positive periodic solution for the system are derived from the intermediate value theorem. We also demonstrate the global stability of the positive periodic solution by the theory of discrete dynamical system. Finally, numerical simulations and discussion are presented to validate our theoretical results.

CONSERVATION OF WILDLIFE. A BIO-ECONOMIC MODEL OF A WILDLIFE RESERVE UNDER THE PRESSURE OF HABITAT DESTRUCTION AND HARVESTING OUTSIDE THE RESERVE

Biodiversity is today threatened by many fac- tors of which destruction and reduction of habitats are con- sidered most important for terrestrial species. One way to counteract these threats is to establish reserves with restric- tions on land use and exploitation. However, very few reserves can be considered islands, wildlife species roam over large ex- panses, often via some density dependent dispersal process. As a consequence, habitat destruction, and exploitation, tak- ing place outside will influence the species abundance inside the conservation area. The paper presents a theoretical model for analyzing this type of management problem. The model presented allows for both the common symmetric dispersal as well as what is called asymmetric dispersal between reserve and outside area. The main finding is that habitat destruction outside may not necessarily have negative impact upon the species abundance in the reserve. As a consequence, economic forces working in the direction of reducing the surrounding habitat have unclear effects on the species abundance within the protected area. We also find that harvesting outside the reserve may have quite modest effect on the species abundance in the reserve. This underlines the attractiveness of reserves from a conservation viewpoint.

Does colonization asymmetry matter in metapopulations?

Despite the considerable evidence showing that dispersal between habitat patches is often asymmetric, most of the metapopulation models assume symmetric dispersal. In this paper, we develop a Monte Carlo simulation model to quantify the effect of asymmetric dispersal on metapopulation persistence. Our results suggest that metapopulation extinctions are more likely when dispersal is asymmetric. Metapopulation viability in systems with symmetric dispersal mirrors results from a mean field approximation, where the system persists if the expected per patch colonization probability exceeds the expected per patch local extinction rate. For asymmetric cases, the mean field approximation underestimates the number of patches necessary for maintaining population persistence. If we use a model assuming symmetric dispersal when dispersal is actually asymmetric, the estimation of metapopulation persistence is wrong in more than 50% of the cases. Metapopulation viability depends on patch connectivity in symmetric systems, whereas in the asymmetric case the number of patches is more important. These results have important implications for managing spatially structured populations, when asymmetric dispersal may occur. Future metapopulation models should account for asymmetric dispersal, while empirical work is needed to quantify the patterns and the consequences of asymmetric dispersal in natural metapopulations.

Dispersal and Dynamics

Mixing and asynchrony of interactions can be expected to stabilize the dynamics of populations. One way such mixing occurs is by dispersal, and Hastings and Gyllenberg et al. have shown that symmetric dispersal between two local populations governed by logistic difference equations can simplify the dynamics. These results are extended here by using a more flexible difference equation and allowing asymmetric dispersal. Although there are some instances where dispersal is destabilizing, its stabilizing effect is enhanced by asymmetry. In addition, very high dispersal rates can induce a stable equilibrium of the metapopulation despite highly chaotic local dynamics. If this equilibrium loses stability, the route to intermittent chaos can be observed. Two new conditions under which dispersal can be stabilizing are discussed. One occurs when the timing of reproduction and dispersal differs in the two patches of the metapopulation. This enlarges the asynchrony of the interactions, and simple dynamics due to dispersal are more likely. The second works by slightly adjusting dispersal rates to control chaotic dynamics. The control can replace chaos by a stable equilibrium. The evolution of dispersal rates is discussed. Since obtaining general criteria for invasion into a population with chaotic dynamics is difficult, no clear conclusions are possible as to whether evolution leads to more stable metapopulations. However, a mutant that controls chaos can invade a resident having the same local dynamics but no control mechanism, so that evolution can lead from chaos to a stable equilibrium.

Mathematical Models of Species Interactions in Time and Space

Three widely used species interaction models, the Nicholson-Bailey and Hassell-Varley host-parasite models and the Lotka-Volterra predator-prey model, are extended into space as well as time. One type of equilibrium having the same number of individuals at all locations is shown to exist analytically for the case. Another kind of equilibrium having different numbers of individuals at different locations in two-dimensional space is demonstrated by simulation. Stability analysis suggests that space-time models having symmetric dispersal are no more stable near equilibrium than the simple time models upon which they are based. In many situations the space-time models are less stable than the simple time models. This result can be reconciled with the experimental literature if one assumes a stochastic persistence effect which predicts that expected extinction time will increase with the number of independent subpopulations.