Effects of convective and dispersive migration on the linear stability of a two species system with mutualistic interactions and functional response

In this paper, effects of convective and dispersive migration on the linear stability of the equilibrium state of a two species system with mutualistic interactions and functional response have been investigated. In both finite and semi-infinite habitats, it has been shown that the otherwise stable equilibrium state without dispersal remains so with dispersal also, both under flux and reservior conditions. In the case of finite habitat, the degree of stability increases as dispersal coefficients of the two species increase. The effect of convective migration also is to stabilize the equilibrium state in this case.     

Stabilization through spatial pattern formation in metapopulations with long-range dispersal

Many studies of metapopulation models assume that spatially extended populations occupy a network of identical habitat patches, each coupled to its nearest neighbouring patches by density-independent dispersal. Much previous work has focused on the temporal stability of spatially homogeneous equilibrium states of the metapopulation, and one of the main predictions of such models is that the stability of equilibrium states in the local patches in the absence of migration determines the stability of spatially homogeneous equilibrium states of the whole metapopulation when migration is added. Here, we present classes of examples in which deviations from the usual assumptions lead to different predictions. In particular, heterogeneity in local habitat quality in combination with long-range dispersal can induce a stable equilibrium for the metapopulation dynamics, even when within-patch processes would produce very complex behaviour in each patch in the absence of migration. Thus, when spatially homogeneous equilibria become unstable, the system can often shift to a different, spatially inhomogeneous steady state. This new global equilibrium is characterized by a standing spatial wave of population abundances. Such standing spatial waves can also be observed in metapopulations consisting of identical habitat patches, i.e. without heterogeneity in patch quality, provided that dispersal is density dependent. Spatial pattern formation after destabilization of spatially homogeneous equilibrium states is well known in reaction–diffusion systems and has been observed in various ecological models. However, these models typically require the presence of at least two species, e.g. a predator and a prey. Our results imply that stabilization through spatial pattern formation can also occur in single-species models. However, the opposite effect of destabilization can also occur: if dispersal is short range, and if there is heterogeneity in patch quality, then the metapopulation dynamics can be chaotic despite the patches having stable equilibrium dynamics when isolated. We conclude that more general metapopulation models than those commonly studied are necessary to fully understand how spatial structure can affect spatial and temporal variation in population abundance.     

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.     

Equilibrium stability of single-species metapopulations

We investigate the effect of migration between local populations of a single discrete-generation species living in a ring or an array of habitats. The commonly used symmetric dispersal assumption is relaxed to include the biologically more reasonable asymmetric dispersion. It is demonstrated analytically that density independent migration has no effect on the equilibrium stability of individual populations. However, the positive equilibrium may be destabilizing if the migration is density dependent in such a way that it increases with increasing population density at the source patch.     

Effects of dispersal on the stability of a gonorrhea endemic model

Using Liapunov's direct method, effects of dispersal on the linear and nonlinear stability of the endemic equilibrium state of the system governing the spread of gonorrhea are investigated. It is noted that the equilibrium state, which is nonlinearly asymptotically stable in the feasible region of the phase plane in the absence of dispersal, remains so with self-dispersal also (cross-dispersal being absent). However, in the presence of both self- and cross-dispersal, the equilibrium state can still remain nonlinearly asymptotically stable in the entire feasible region provided a certain condition involving self- and cross-dispersal coefficients is satisfied. It is also seen in this case that, for the linearly stable equilibrium state, there exists a subregion of the feasible region where it is nonlinearly asymptotically stable.     

Evolution of dispersal in density regulated populations: A haploid model

The evolution of dispersal is explored in a density-dependent framework. Attention is restricted to haploid populations in which the genotypic fitnesses at a single diallelic locus are decreasing functions of the changing number of individuals in the population. It is shown that migration between two populations in which the genotypic response to density is reversed can maintain both alleles when the intermigration rates are constant or nondecreasing functions of the population densities. There is always a unique symmetric interior equilibrium with equal numbers but opposite gene frequencies in the two populations, provided the system is not degenerate. Numerical examples with exponential and hyperbolic fitnesses suggest that this is the only stable equilibrium state under constant positive migration rates (m) less than . Practically speaking, however, there is only convergence after a reasonable number of generations for relatively small migration rates ( ). A migration-modifying mutant at a second, neutral locus, can successfully enter two populations at a stable migration-selection balance if and only if it reduces the intermigration rates of its carriers at the original equilibrium population size. Moreover, migration modification will always result in a higher equilibrium population size, provided the system approaches another symmetric interior equilibrium. The new equilibrium migration rate will be lower than that at the original equilibrium, even when the modified migration rate is a nondecreasing function of the population sizes. Therefore, as in constant viability models, evolution will lead to reduced dispersal.     

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.     

Stability in an age-structured metapopulation model

We present a discrete model for a metapopulation of a single species with overlapping generations. Based on the dynamical behavior of the system in absence of dispersal, we have shown that a migration mechanism which depends only on age can not stabilize a previously unstable homogeneous equilibrium, but can drive a stable uncoupled system to instability if the migration rules are strongly related to age structure.     

Effects of convective and dispersive interactions on the stability of two species

The evolution and local stability of a system of two interacting species in a finite two-dimensional habitat is investigated by taking into account the effects of self- and cross-dispersion and convection of the species. In absence of cross-dispersion, an equilibrium state which is stable without dispersion is always stable with dispersion provided that the dispersion coefficients of the two species are equal. However, when the dispersion coefficients of the two species are different, the possibility of self-dispersive instability arises. It is also pointed out that the cross-dispersion of species may lead to stability or instability depending upon the nature and the magnitude of the cross-dispersive interactions in comparison to the self-dispersive interactions. The self-convective movement of species increases the stability of the equilibrium state and can stabilize an otherwise unstable equilibrium state. The effect of cross-convection (in absence of self-dispersion and self-convection) is to stabilize the equilibrium state in a prey-predator model with positive cross-dispersion coefficients for the prey species. Finally, it is shown that if the system is stable under homogeneous boundary conditions it remains so under non-homogeneous boundary conditions.     

Sex-specific genetic differentiation and coalescence times: estimating sex-biased dispersal rates

I derive the equilibrium values of sex-specific FST parameters, in an island model for a dioecious species with sex-biased dispersal and binomial distribution of family size before dispersal (as assumed in a Wright-Fisher population). I show that FST may take different values among males and among females whenever dispersal is a trait conditioned on gender. This has not always been recognized, because some models assumed that genes are sampled before dispersal. In particular, the ratios of sex-specific FST parameters evaluated after dispersal over FST evaluated before dispersal are simple functions of sex-specific dispersal rates. Therefore, a simple moment-based estimator of sex-specific dispersal rate is proposed. This method is based on the comparison of FST estimated before and after dispersal and assumes equilibrium between migration and drift. I evaluate this method through stochastic simulations for a range of sex-specific dispersal rates and sampling effort (sample size, number of loci scored).     

Habitat stability affects dispersal and the ability to track climate change

Habitat persistence should influence dispersal ability, selecting for stronger dispersal in habitats of lower temporal stability. As standing (lentic) freshwater habitats are on average less persistent over time than running (lotic) habitats, lentic species should show higher dispersal abilities than lotic species. Assuming that climate is an important determinant of species distributions, we hypothesize that lentic species should have distributions that are closer to equilibrium with current climate, and should more rapidly track climatic changes. We tested these hypotheses using datasets from 1988 and 2006 containing all European dragon- and damselfly species. Bioclimatic envelope models showed that lentic species were closer to climatic equilibrium than lotic species. Furthermore, the models over-predicted lotic species ranges more strongly than lentic species ranges, indicating that lentic species track climatic changes more rapidly than lotic species. These results are consistent with the proposed hypothesis that habitat persistence affects the evolution of dispersal.     

Dispersal delays, predator-prey stability, and the paradox of enrichment

It takes time for individuals to move from place to place. This travel time can be incorporated into metapopulation models via a delay in the interpatch migration term. Such a term has been shown to stabilize the positive equilibrium of the classical Lotka-Volterra predator-prey system with one species (either the predator or the prey) dispersing. We study a more realistic, Rosenzweig-MacArthur, model that includes a carrying capacity for the prey, and saturating functional response for the predator. We show that dispersal delays can stabilize the predator-prey equilibrium point despite the presence of a Type II functional response that is known to be destabilizing. We also show that dispersal delays reduce the amplitude of oscillations when the equilibrium is unstable, and therefore may help resolve the paradox of enrichment.     

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.     

The ideal free distribution as an evolutionarily stable strategy

We examine the evolutionary stability of strategies for dispersal in heterogeneous patchy environments or for switching between discrete states (e.g. defended and undefended) in the context of models for population dynamics or species interactions in either continuous or discrete time. There have been a number of theoretical studies that support the view that in spatially heterogeneous but temporally constant environments there will be selection against unconditional, i.e. random, dispersal, but there may be selection for certain types of dispersal that are conditional in the sense that dispersal rates depend on environmental factors. A particular type of dispersal strategy that has been shown to be evolutionarily stable in some settings is balanced dispersal, in which the equilibrium densities of organisms on each patch are the same whether there is dispersal or not. Balanced dispersal leads to a population distribution that is ideal free in the sense that at equilibrium all individuals have the same fitness and there is no net movement of individuals between patches or states. We find that under rather general assumptions about the underlying population dynamics or species interactions, only such ideal free strategies can be evolutionarily stable. Under somewhat more restrictive assumptions (but still in considerable generality), we show that ideal free strategies are indeed evolutionarily stable. Our main mathematical approach is invasibility analysis using methods from the theory of ordinary differential equations and nonnegative matrices. Our analysis unifies and extends previous results on the evolutionary stability of dispersal or state-switching strategies.     

Effects of local interactions and local migration on stability

Mathematical models can help to resolve the longstanding question of whether more diverse communities are more stable. Here, I focus on how local dispersal and local interactions--hallmarks of spatial communities--affect stability in a spatially implicit model with demographic stochasticity. The results are based on a novel way to analyze moment equations. The main conclusion is that the type and strength of density-dependent factors, such as fecundity and competition, determine whether local dispersal and local interactions increase or decrease stability. Local dispersal has a stabilizing effect when fecundity is high, interspecific competition is either low or high, and the number of species is small. Effects of local migration on stability are amplified when space is explicit.     

Effects of diffusion on the stability of the equilibrium in multi-species ecological systems

Continuous population distributions that undergo self-diffusion, migrational cross-diffusion and interaction in a region of (1-, 2- or 3-dimensional) space are described dynamically by a governing system of nonlinear reaction-diffusion equations. It is shown that the constants associated with migrational cross-diffusion are ordinarily nonnegative or nonpositive, contingent on the type of species interaction. A simple sign relationship obtains between the latter diffusivity constants and the rate constants for species interaction in the neighborhood of a spatially uniform equilibrium state, and this relationship of signs serves to simplify the general stability theory for the growth or decay of arbitrary perturbations on a spatially uniform equilibrium state. The stability of the equilibrium state is analyzed and discussed in detail for the case of a generic two-species model, where the self-diffusion and migrational cross-diffusion of species act to either stabilize or destabilize the equilibrium, depending essentially on the character of the species interaction and also on the magnitude of the Helmholtz eigenvalues associated with the region and boundary conditions. In particular, for a prey-predator or host-parasite model, self-diffusion usually helps to stabilize the equilibrium state and migrational cross-diffusion can only act as an additional stabilizing influence, as evidenced generally by the experiments on such two-species systems. Sufficient conditions are derived for stability of the equilibrium state in the general case for an arbitrarily large number of interacting species. It is shown that the equilibrium state is always stable if all species undergo significant self-diffusion and the Helmholtz eigenvalues are suitably large.     

Evolutionary Convergence to Ideal Free Dispersal Strategies and Coexistence

We study a two species competition model in which the species have the same population dynamics but different dispersal strategies and show how these dispersal strategies evolve. We introduce a general dispersal strategy which can result in the ideal free distributions of both competing species at equilibrium and generalize the result of Averill et al. (2011). We further investigate the convergent stability of this ideal free dispersal strategy by varying random dispersal rates, advection rates, or both of these two parameters simultaneously. For monotone resource functions, our analysis reveals that among two similar dispersal strategies, selection generally prefers the strategy which is closer to the ideal free dispersal strategy. For nonmonotone resource functions, our findings suggest that there may exist some dispersal strategies which are not ideal free, but could be locally evolutionarily stable and/or convergent stable, and allow for the coexistence of more than one species.     

Are Plant Species Able to Keep Pace with the Rapidly Changing Climate?

Future climate change is predicted to advance faster than the postglacial warming. Migration may therefore become a key driver for future development of biodiversity and ecosystem functioning. For 140 European plant species we computed past range shifts since the last glacial maximum and future range shifts for a variety of Intergovernmental Panel on Climate Change (IPCC) scenarios and global circulation models (GCMs). Range shift rates were estimated by means of species distribution modelling (SDM). With process-based seed dispersal models we estimated species-specific migration rates for 27 dispersal modes addressing dispersal by wind (anemochory) for different wind conditions, as well as dispersal by mammals (dispersal on animal''s coat – epizoochory and dispersal by animals after feeding and digestion – endozoochory) considering different animal species. Our process-based modelled migration rates generally exceeded the postglacial range shift rates indicating that the process-based models we used are capable of predicting migration rates that are in accordance with realized past migration. For most of the considered species, the modelled migration rates were considerably lower than the expected future climate change induced range shift rates. This implies that most plant species will not entirely be able to follow future climate-change-induced range shifts due to dispersal limitation. Animals with large day- and home-ranges are highly important for achieving high migration rates for many plant species, whereas anemochory is relevant for only few species.     

Coupling of two competitive systems via density dependent migration

Coupling of two Lotka–Volterra type competition systems with density-dependent migration was surveyed. We assumed that species x and y are each exclusively superior in subhabitat 1 and subhabitat 2, respectively, and that population densities that exert intra-and interspecific competitive effects also impose pressures for migration of individuals from a subhabitat. If the two species are, respectively, abundant in the subhabitats in which either species is competitively superior, and the migration has a mixing effect, then, it would be intuitively expected that, as potential migration rates increase, the two species are mixed well and coexist in the whole habitat. An analysis of this competitive situation using our model under the assumption of linear diffusion predicted that, even though weak mixing maintains coexistence in the whole habitat, strong mixing collapses coexistence and leads to the exclusion of one species. The assumption that migrations occur due to self- and cross-population pressures provides different predictions: (i) weak dominance and strong mixing destabilize the coexistence state and lead to a monopolizing equilibrium of either species (bi-stability of monopolizing equiliblia); (ii) conspicuous weakness of the inferior species makes the mixing equilibrium stable, regardless of the potential migration rate; and (iii) tri-stability exists in between situations (i) and (ii). In the third case, the attainable state is the mixing equilibrium or either of the monopolizing equilibria, depending on the initial state. Migration mechanisms with self- and cross-population pressures tends to mediate spatial segregation and makes coexistence possible, even with strong mixing.     

The ideal free distribution as an evolutionarily stable strategy

We examine the evolutionary stability of strategies for dispersal in heterogeneous patchy environments or for switching between discrete states (e.g. defended and undefended) in the context of models for population dynamics or species interactions in either continuous or discrete time. There have been a number of theoretical studies that support the view that in spatially heterogeneous but temporally constant environments there will be selection against unconditional, i.e. random, dispersal, but there may be selection for certain types of dispersal that are conditional in the sense that dispersal rates depend on environmental factors. A particular type of dispersal strategy that has been shown to be evolutionarily stable in some settings is balanced dispersal, in which the equilibrium densities of organisms on each patch are the same whether there is dispersal or not. Balanced dispersal leads to a population distribution that is ideal free in the sense that at equilibrium all individuals have the same fitness and there is no net movement of individuals between patches or states. We find that under rather general assumptions about the underlying population dynamics or species interactions, only such ideal free strategies can be evolutionarily stable. Under somewhat more restrictive assumptions (but still in considerable generality), we show that ideal free strategies are indeed evolutionarily stable. Our main mathematical approach is invasibility analysis using methods from the theory of ordinary differential equations and nonnegative matrices. Our analysis unifies and extends previous results on the evolutionary stability of dispersal or state-switching strategies.