Effects of Immigration on the Dynamics of Simple Population Models

Many simple population models exhibit the period doubling route to chaos as a single parameter, commonly the growth rate, is increased. Here we examine the effect of an immigration process on such models and explain why in the case of one-dimensional ("single-humped") maps, immigration often tends to suppress chaos and stabilise equilibrium behaviour or cyclical oscillations of long period. The conditions for which an increase of immigration "simplifies" population dynamics are examined.     

On linear perturbations of the Ricker model

A class of linearly perturbed discrete-time single species scramble competition models, like the Ricker map, is considered. Perturbations can be of both recruitment and harvesting types. Stability (bistability) is considered for models, where parameters of the map do not depend on time. For models with recruitment, the result is in accordance with Levin and May conjecture [S.A. Levin, R.M. May, A note on difference delay equations, Theor. Pop. Biol. 9 (1976) 178]: the local stability of the positive equilibrium implies its global stability. For intrinsic growth rate r-->infinity the way to chaos is broken down to get extinction of population for the depletion case and to establish a stable two-cycle period for models with immigration. The latter behaviour is also studied for models with random discrete constant perturbations of recruitment type. Extinction, persistence and existence of periodic solutions are studied for the perturbed Ricker model with time-dependent parameters.     

Incorporating spatial variation in density enhances the stability of simple population dynamics models

Simple discrete time models of population growth admit a wide variety of dynamic behaviors, including population cycles and chaos. Yet studies of natural and laboratory populations typically reveal their dynamics to be relatively stable. Many explanations for the apparent rarity of unstable or chaotic behavior in real populations have been developed, including the possible stabilizing roles of migration, refugia, abrupt density-dependence, and genetic variation in sensitivity to density. We develop a theoretical framework for incorporating random spatial variation in density into simple models of population growth, and apply this approach to two commonly used models in ecology: the Ricker and Hassell maps. We show that the incorporation of spatial density variation into both these models has a strong stabilizing influence on their dynamic behavior, and leads to their exhibiting stable point equilibria or stable limit cycles over a relatively much larger range of parameter values. We suggest that one reason why chaotic population dynamics are less common than the simple models indicate is, these models typically neglect the potentially stabilizing role of spatial variation in density.     

Behaviour of simple population models under ecological processes

The two most popular and extensively-used discrete models of population growth display the generic bifurcation structure of a hierarchy of period-doubling sequence to chaos with increasing growth rates. In this paper we show that these two models, though they belong to a general class of one-dimensional maps, show very different dynamics when important ecological processes such as immigration and emigration/depletion, are considered. It is important that ecologists recognize the differences between these models before using them to describe their data—or develop optimization strategies—based on these models.     

Contributions of adult oligochaete emigration and immigration in a dynamic soft-sediment community

It is well known that adult dispersal is common in soft bottom intertidal and shallow subtidal communities. We here report on the first study that attempts to quantify the effects of both immigration and emigration on patches of soft sediment communities. Some species show adaptive emigration from the seabed, although dispersal direction, distance, and colonization success are probably strongly dependent on hydrodynamics, morphological adaptations to dispersal, and the ability to select appropriate target microsites. The naid oligochaete Paranais litoralis is a numerically dominant benthic species in southern New England and New York mud flats and tends to reproduce mainly or exclusively by means of budding of new individuals. When population density is high and resources in short supply, budding frequency is reduced, worms grow longer, and may emigrate from the sediment. We quantified emigration by means of a conical trap and quantified immigration with sediment dishes. We followed emigration/immigration during the typical late spring population explosion and crash cycle of worms within the sediment, which is driven by a seasonal cycle of provision and exhaustion of organic detrital food supply. Emigration was proportionally maximal either at or after the population peak, consistent with a response to food shortage. Over a span of ca. 50 m, we found no net movement in either direction along a transect, nor was emigration or immigration correlated with local density in the sediment. Nevertheless, both emigration and immigration were important in our 2004 sampling, and immigration especially had an important impact on population densities. We do not know the relative capture efficiencies of the emigration and immigration apparatus, so more needs to be done to understand the impacts of dispersal in this and other systems.     

Finite metapopulation models with density-dependent migration and stochastic local dynamics

The effects of small density-dependent migration on the dynamics of a metapopulation are studied in a model with stochastic local dynamics. We use a diffusion approximation to study how changes in the migration rate and habitat occupancy affect the rates of local colonization and extinction. If the emigration rate increases or if the immigration rate decreases with local population size, a positive expected rate of change in habitat occupancy is found for a greater range of habitat occupancies than when the migration is density-independent. In contrast, the reverse patterns of density dependence in respective emigration and immigration reduce the range of habitat occupancies where the metapopulation will be viable. This occurs because density-dependent migration strongly influences both the establishment and rescue effects in the local dynamics of metapopulations.     

Metapopulation structure and habitat quality in modelling dispersal in the butterfly Iolana iolas

Quantifying dispersal is fundamental to understanding the effects of fragmentation on populations. Although it has been shown that patch and matrix quality can affect dispersal patterns, standard metapopulation models are usually based on the two basic variables, patch area and connectivity. In 2004 we studied migration patterns among 18 habitat patches in central Spain for the butterfly Iolana iolas, using mark–release–recapture methods. We applied the virtual migration (VM) model and estimated the parameters of emigration, immigration and mortality separately for males and females. During parameter estimation and model simulations, we used original and modified patch areas accounting for habitat quality with three different indices. Two indices were based on adult and larval resources (flowers and fruits) and the other one on butterfly density. Based on unmodified areas, our results showed that both sexes were markedly different in their movements and mortality rates. Females emigrated more frequently from patches, but males that emigrated were estimated to move longer daily dispersal distances and suffer higher mortality than females during migration. Males were more likely to emigrate from small than from large patches, but patch area had no significant effect on female emigration. The effects of area on immigration rate and the within-patch mortality were similar in both sexes. Based on modified areas, the estimated parameter values and the model simulation results were similar to those estimated using the unmodified patch areas. One possible reason for the failure to significantly improve the parameter estimates of the VM model is the fact that resource quantity and butterfly population sizes were strongly correlated with patch area. Our results suggest that the standard VM modelling approach, based on patch area and connectivity, can provide a realistic picture of the movement patterns of I. iolas .     

Source populations in carnivore management: cougar demography and emigration in a lightly hunted population

Wildlife agencies typically attempt to manage carnivore numbers in localized game management units through hunting, and do not always consider the potential influences of immigration and emigration on the outcome of those hunting practices. However, such a closed population structure may not be an appropriate model for management of carnivore populations where immigration and emigration are important population parameters. The closed population hypothesis predicts that high hunting mortality will reduce numbers and densities of carnivores and that low hunting mortality will increase numbers and densities. By contrast, the open population hypothesis predicts that high hunting mortality may not reduce carnivore densities because of compensatory immigration, and low hunting mortality may not result in more carnivores because of compensatory emigration. Previous research supported the open population hypothesis with high immigration rates in a heavily hunted (hunting mortality rate=0.24) cougar population in northern Washington. We test the open population hypothesis and high emigration rates in a lightly hunted (hunting mortality rate=0.11) cougar population in central Washington by monitoring demography from 2002 to 2007. We used a dual sex survival/fecundity Leslie matrix to estimate closed population growth and annual census counts to estimate open population growth. The observed open population growth rate of 0.98 was lower than the closed survival/fecundity growth rates of 1.13 (deterministic) and 1.10 (stochastic), and suggests a 12–15% annual emigration rate. Our data support the open population hypothesis for lightly hunted populations of carnivores. Low hunting mortality did not result in increased numbers and densities of cougars, as commonly believed because of compensatory emigration.     

Evaluation of alternate harvesting strategies using experimental microcosms

Experimental evidence to evaluate alternate conservation policies for harvested populations is currently meager. We used populations of the ciliate Tetrahymena thermophila growing in test tube microcosms to experimentally evaluate the effects of alternate harvesting policies in a controlled, replicable setting. Simple density-dependent models were effective in predicting patterns of ciliate population growth in the microcosms. We evaluated several univariate models, finding that a Ricker logistic model was a better predictor of ciliate population dynamics than Gompertz logistic, non-linear logistic, or random walk models. Using the Ricker logistic model as a demographic skeleton, we modeled ciliate population dynamics with respect to three alternate harvesting policies (fixed quota, fixed proportion, and fixed escapement), each conducted at four comparable levels of harvest intensity. The parameterized demographic models predicted that fixed quota harvesting would lead to lower mean ciliate abundance and higher temporal variability in ciliate abundance than fixed proportion or fixed escapement policies, with an appreciable risk of extinction, even under the controlled environmental conditions of our experimental system. For each harvesting policy, the intensity of harvest had demonstrable effects on population density. Population variability was higher for fixed quota harvesting than the other policies. The stochastic demographic model successfully predicted heightened extinction risk in the fixed quota system, relative to the other management treatments. Our experimental evidence lends support to the theoretical prediction that fixed quota harvesting is riskier than fixed proportion or fixed escapement policies.     

Relating individual behaviour to population dynamics

How do the behavioural interactions between individuals in an ecological system produce the global population dynamics of that system? We present a stochastic individual-based model of the reproductive cycle of the mite Varroa jacobsoni, a parasite of honeybees. The model has the interesting property in that its population level behaviour is approximated extremely accurately by the exponential logistic equation or Ricker map. We demonstrated how this approximation is obtained mathematically and how the parameters of the exponential logistic equation can be written in terms of the parameters of the individual-based model. Our procedure demonstrates, in at least one case, how study of animal ecology at an individual level can be used to derive global models which predict population change over time.     

迁移对害虫抗性演化的影响

利用抗性群体遗传模型研究了迁移对害虫抗性演化的影响。模拟结果表明,迁入比例对害虫种群抗性演化的影响存在一定的阈值。若迁入个体全为敏感(SS)纯合子,而其比例又在阈值以下时,这种迁入对抗性演化无延缓作用;迁入比例在阈值以上时,随比例加大,对抗性延缓作用逐渐增强,加大到某一值时,抗性演化可完全被阻止;若迁入个体携有R基因,则迁入比例在阈值以下时,会促进迁入区害虫种群的抗性演化;在阈值以上时,可延缓或完全阻止害虫种群的抗性演化。处理区害虫种群原有个体的迁出,可加强SS个体迁入对害虫种群抗性演化的影响,而使携R基因的个体的迁入影响减弱。以小菜蛾Plutella xylostella为模型昆虫所作的试验结果表明,模型模拟与验证试验的结果具有较好的一致性,说明抗性模拟模型可用于害虫种群的抗性演化预测和分析。     

A comparative evaluation of models of cinnabar moth dynamics

Summary 1. A complex model of cinnabar moth dynamics proposed by Dempster and Lakhani (1979) with 23 parameters is reduced to a single equation with five parameters, and the behaviour of the reduced model shown to explain most features of the full model. 2. The efficiency of the full model is compared with the reduced model and with two even simpler models (the two parameter discrete logistic and a four parameter model based on a step-function for mortality) in their abilities to describe time series data of cinnabar moth population densities from Weeting Heath. Models with more parameters were not significantly better than few-parameter models in describing population trajectories. 3. Models that included a driving variable (in this case observed rainfall data) were no better at describing the data than simpler models without driving variables. It appears, therefore, that the routine inclusion of driving variables may be counterproductive, unless there is compelling empirical or theoretical evidence of their importance and the mode of action of the driving variables can be modelled mechanistically. For example, the regression model used to describe the relationship between rainfall and plant biomass in Dempster and Lakhani (1979), breaks down if rainfall is assumed to be constant, because there is no explicit model for the regulation of plant biomass. 4. The parameter values of the cinnabar-ragwort interaction suggest that cinnabar moth dynamics may be chaotic. Whether or not field data exhibit chaos or environmental stochasticity (or a mixture of both) is impossible to determine from inspection of time series data on population density. There is an urgent need for experimental and theoretical protocols to disentangle these two sources of population fluctuation.     

On the evolution of specialization with a mechanistic underpinning in structured metapopulations

We analyze the evolution of specialization in resource utilization in a discrete-time metapopulation model using the adaptive dynamics approach. The local dynamics in the metapopulation are based on the Beverton-Holt model with mechanistic underpinnings. The consumer faces a trade-off in the abilities to consume two resources that are spatially heterogeneously distributed to patches that are prone to local catastrophes. We explore the factors favoring the spread of generalist or specialist strategies. Increasing fecundity or decreasing catastrophe probability favors the spread of the generalist strategy and increasing environmental heterogeneity enlarges the parameter domain where the evolutionary branching is possible. When there are no catastrophes, increasing emigration diminishes the parameter domain where the evolutionary branching may occur. Otherwise, the effect of emigration on evolutionary dynamics is non-monotonous: both small and large values of emigration probability favor the spread of the specialist strategies whereas the parameter domain where evolutionary branching may occur is largest when the emigration probability has intermediate values. We compare how different forms of spatial heterogeneity and different models of local growth affect the evolutionary dynamics. We show that even small changes in the resource dynamics may have outstanding evolutionary effects to the consumers.     

Concordant genetic estimators of migration reveal anthropogenically enhanced source-sink population structure in the river sculpin, Cottus gobio

River systems are vulnerable to natural and anthropogenic habitat fragmentation and will often harbor populations deviating markedly from simplified theoretical models. We investigated fine-scale population structure in the sedentary river fish Cottus gobio using microsatellites and compared migration estimates from three FST estimators, a coalescent maximum-likelihood method and Bayesian recent migration analyses. Source-sink structure was evident via asymmetry in migration and genetic diversity with smaller upstream locations emigration biased and larger downstream subpopulations immigration biased. Patterns of isolation by distance suggested that the system was largely, but not entirely, in migration-drift equilibrium, with headwater populations harboring a signal of past colonizations and in some cases also recent population bottlenecks. Up- vs. downstream asymmetry in population structure was partly attributable to the effects of flow direction, but was enhanced by weirs prohibiting compensatory upstream migration. Estimators of migration showed strong correspondence, at least in relative terms, especially if pairwise FST was used as an indirect index of relative gene flow rather than being translated to Nm. Since true parameter values are unknown in natural systems, comparisons among estimators are important, both to determine confidence in estimates of migration and to validate the performance of different methods.     

Influence of food availability on demography and local population dynamics in a long-lived seabird

Few studies have addressed the effects of food availability as a proximate factor affecting local adult survival in long-lived organisms and their consequences at local population dynamics. We used capture-recapture analysis of resightings of 10 birth cohorts of ringed Audouin's gulls, Larus audouinii, to estimate adult survival and dispersal (both emigration and immigration). For the first time, permanent emigration (the transient effect in capture-recapture analysis) was modelled for the whole population and not only for the newly marked birds. Gulls exploit to a large extent fishes discarded from trawlers, and a trawling moratorium established since 1991 has decreased food supply for the colony. This was used as a natural experiment of food availability to assess its effects on adult survival and emigration. These and other demographic parameters were used in a projection modelling to assess the probabilities of extinction of the colony under two scenarios of lower and higher food availability. Food availability (together with the age of individuals) influenced emigration probabilities, but not adult survival, which was estimated at 0.91 (s.e. = 0.02). When food was in shorter supply during the chick-rearing period, emigration was very high (ca. 65%) for younger breeders, although this rate decreased sharply with age. Probabilities of extinction were very high when food availability was low, and when environmental stochasticity was introduced, and only stochastic immigration from the outside seemed to prevent extinction. The results highlight the importance of dispersal processes in the population dynamics of long-lived organisms.     

Effects of immigration on some stochastic logistic models: a cumulant truncation analysis.

This paper uses a new cumulant truncation methodology to investigate the stochastic power law logistic model with immigration, and illustrates the model with parameter values used to describe the growth of muskrat populations in the Netherlands. This model has a stable equilibrium distribution. The incorporation of immigration into the model, therefore, simplifies the qualitative nature of the stochastic solution. The (unconditional) cumulant functions for the transient and the equilibrium population size distributions are obtained, from which the distributions are shown to be near-normal at all times for the parameter values of interest. Approximating cumulant functions, which are relatively easy to find in practice, are derived and shown to be quite accurate, except for the case of massive immigration. As the level of immigration increases, the mean value rises more rapidly initially, as expected; however, the variance and the skewness of both the transient and the equilibrium distributions are reduced.     

Dynamics of mutualist populations that are demographically open

1. Few theoretical studies have examined the impact of immigration and emigration on mutualist population dynamics, but a recent empirical study (A.R. Thompson Oecologia, 143, 61-69) on mutualistic fish and shrimp showed that immigration can prevent population collapse, and that intraspecific competition for a mutualistic partner can curb population expansion. To understand in a theoretical context the implications of these results, and to assess their generality, we present a two-species model that accounts explicitly for immigration and emigration, as well as distinguishing the impacts of mutualism on birth rates, death rates and habitat acquisition. 2. The model confirms that immigration can stabilize mutualistic populations, and predicts that high immigration, along with enhanced reproduction and/or reduced mortality through mutualism, can cause population sizes to increase until habitat availability curbs further expansion. 3. We explore in detail the effects of different forms of habitat limitation on mutualistic populations. Habitat availability commonly limits the density of both populations if mutualists acquire shelter independently. If a mutualist depends on a partner for habitat, densities of that mutualist are capped by the amount of space provided by that partner. The density of the shelter-provider is limited by the environment. 4. If a mutualism solely augments reproduction, and most locally produced individuals leave the focal patch, then the mutualism will have a minimal effect on local dynamics. If the mutualism operates by reducing rates of death or enhancing habitat availability, and there is at least some immigration, then mutualism will affect local dynamics. This finding may be particularly relevant in marine systems, where there is high variability (among species and locations) in the extent to which progeny disperse from natal locations. 5. Overall, our results demonstrate that the consequences of immigration and emigration for the dynamics of mutualists depend strongly on which demographic rate is influenced by mutualism. 6. By relating our model to a variety of terrestrial and aquatic systems, we provide a general framework to guide future empirical studies of the dynamics of mutualistic populations.     

Immigration can destabilize tri-trophic interactions: implications for conservation of top predators

Top predators often have large home ranges and thus are especially vulnerable to habitat loss and fragmentation. Increasing connectance among habitat patches is therefore a common conservation strategy, based in part on models showing that increased migration between subpopulations can reduce vulnerability arising from population isolation. Although three-dimensional models are appropriate for exploring consequences to top predators, the effects of immigration on tri-trophic interactions have rarely been considered. To explore the effects of immigration on the equilibrium abundances of top predators, we studied the effects of immigration in the three-dimensional Rosenzweig-MacArthur model. To investigate the stability of the top predator equilibrium, we used MATCONT to perform a bifurcation analysis. For some combinations of model parameters with low rates of top predator immigration, population trajectories spiral towards a stable focus. Holding other parameters constant, as immigration rate is increased, a supercritical Hopf bifurcation results in a stable limit cycle and thus top predator populations that cycle between high and low abundances. Furthermore, bistability arises as immigration of the intermediate predator is increased. In this case, top predators may exist at relatively low abundances while prey become extinct, or for other initial conditions, the relatively higher top predator abundance controls intermediate predators allowing for non-zero prey population abundance and increased diversity. Thus, our results reveal one of two outcomes when immigration is added to the model. First, over some range of top predator immigration rates, population abundance cycles between high and low values, making extinction from the trough of such cycles more likely than otherwise. Second, for relatively higher intermediate predator migration rates, top predators may exist at low values in a truncated system with impoverished diversity, again with extinction more likely.