Combining Perturbations and Parameter Variation to Influence Mean First Passage Times

Perturbations are relatively large shocks to state variables that can drive transitions between stable states, while drift in parameter values gradually alters equilibrium magnitudes. This latter effect can lead to equilibrium bifurcation, the generation, or annihilation of equilibria. Equilibrium annihilations reduce the number of equilibria and so are associated with catastrophic population collapse. We study the combination of perturbations and parameter drift, using a two-species intraguild predation (IGP) model. For example, we use bifurcation analysis to understand how parameter drift affects equilibrium number, showing that both competition and predation rates in this model are bifurcating parameters. We then introduce a stochastic process to model the effects of population perturbations. We demonstrate how to evaluate the joint effects of perturbations and drift using the common currency of mean first passage time to transitions between stable states. Our methods and results are quite general, and for example, can relate to issues in both pest control and sustainable harvest. Our results show that parameter drift (1) does not importantly change the expected time to reach target points within a basin of attraction, but (2) can dramatically change the expected time to shift between basins of attraction, through its effects on equilibrium resilience.     

Continuous-discrete models of the dynamics of an isolated population and of 2 competing species

The paper presents the analysis of various mathematical models for dynamics of isolated population and for competition between two species. It is assumed that mortality is continuous and birth of individuals of new generations takes place in certain fixed moments. Influence of winter upon the population dynamics and conditions of classic discrete model "deduction" of population dynamics (in particular, Moran-Ricker and Hassel's models) are investigated. Dynamic regimes of models under various assumptions about the birth and death rates upon the population states are also examined. Analysis of models of isolated population dynamics with nonoverlapping generations showed the density changes regularly if the birth rate is constant. Moreover, there exists a unique global stable level and population size stabilizes asymptotically at this equilibrium, i.e. cycle and chaotic regimes in various discrete models depend on correlation between individual productivity and population state in previous time. When the correlation is exponential upon mean population size the discrete Hassel model is realized. Modification of basis model, based on the assumption that during winter survival/death changes are constant, showed that population size at global level is stable. Generally, the dependence of population rate upon "winter parameters" has nonlinear character. Nonparametric models of competition between two species does not vary if the individual productivity is constant. In a phase space there are several stable stationary states and population stabilizes at one or other level asymptotically. So, in discrete models of competition between two species oscillation can be explained by dependence of population growth rate on the population size at previous times.     

Windborne displacements of Desert Locusts from Africa to the Caribbean and South America

The Desert Locust is a major pest of agriculture in Africa, the Middle East and South-West Asia and swarms are known to make downwind flights over hundreds and thousands of kilometres between seasonal breeding areas. At the end of summer in 1988, swarms of locusts were moving north and south along the western margins of North Africa and in October and November, swarms crossed the Atlantic Ocean and invaded the Caribbean and neighbouring parts of South America for the first recorded time. Because of the extent of the migration and the evolutionary significance of linkages between Old and New World species of locusts, the weather associated with the migrations was studied and trajectory analysis was used to identify the source areas and estimate the flight times. Locusts were moving offshore from western North Africa throughout the autumn and on three occasions migrated west of 40° W with easterly Trade winds. Two trans-Atlantic crossings coincided with the passage of easterly waves. Over 100 trajectories were constructed at 950 and 850 hPa and within the time limit used ( 144-h), 28% successfully linked source and receptor areas. Minimum trajectory duration was 93-h, which is one-and-a-half times longer than the previously longest flight duration, derived for a similar migration to the British Isles in 1954. Upwind trajectories from the arrival areas, identified sources between 27 and 6° N in Africa, with most end-points located in Mauritania, Senegal, Gambia and Guinea-Bissau. Interspersed with the Atlantic crossings were a northward movement of locusts and an incursion of Saharan dust into Europe within the circulations of frontal depressions. While offshore migrations from northern Africa are common in autumn, the immigrants in the Caribbean and South America were probably at the extreme limits of flight endurance for the species. The results tend to confirm earlier hypotheses that New World species of locusts may have evolved from ancestral migrants from Africa.     

Pest population stability under sterile releases

The stability of a pest population is one of the critical features to be examined when considering a control strategy for a given pest species. Four models involviing sterile male releases are examined for various stability characteristics; the models examined were: (i) a simple one stage model with no species interactions, (ii) a two life-stage model, (iii) a model involving two competing species, (iv) a model in which the pest is under predation. Of the four, the simple model was the most stable and the predation model was the least stable under continued sterile releases.     

The maintenance (or not) of polygenic variation by soft selection in heterogeneous environments

On the basis of single-locus models, spatial heterogeneity of the environment coupled with strong population regulation within each habitat (soft selection) is considered an important mechanism maintaining genetic variation. We studied the capacity of soft selection to maintain polygenic variation for a trait determined by several additive loci, selected in opposite directions in two habitats connected by dispersal. We found three main types of stable equilibria. Extreme equilibria are characterized by extreme specialization to one habitat and loss of polymorphism. They are analogous to monomorphic equilibria in singe-locus models and are favored by similar factors: high dispersal, weak selection, and low marginal average fitness of intermediate genotypes. At the remaining two types of equilibria the population mean is intermediate but variance is very different. At fully polymorphic equilibria all loci are polymorphic, whereas at low-variance equilibria at most one locus remains polymorphic. For most parameters only one type of equilibrium is stable; the transition between the domains of fully polymorphic and low-variance equilibria is typically sharp. Low-variance equilibria are favored by high marginal average fitness of intermediate genotypes, in contrast to single-locus models, in which marginal overdominance is particularly favorable for maintenance of polymorphism. The capacity of soft selection to maintain polygenic variation is thus more limited than extrapolation from single-locus models would suggest, in particular if dispersal is high and selection weak. This is because in a polygenic model, variance can evolve independently of the mean, whereas in the single-locus two-allele case, selection for an intermediate mean automatically leads to maintenance of polymorphism.     

Evolutionary dynamics and strong Allee effects

We describe the dynamics of an evolutionary model for a population subject to a strong Allee effect. The model assumes that the carrying capacity k(u), inherent growth rate r(u), and Allee threshold a(u) are functions of a mean phenotypic trait u subject to evolution. The model is a plane autonomous system that describes the coupled population and mean trait dynamics. We show bounded orbits equilibrate and that the Allee basin shrinks (and can even disappear) as a result of evolution. We also show that stable non-extinction equilibria occur at the local maxima of k(u) and that stable extinction equilibria occur at local minima of r(u). We give examples that illustrate these results and demonstrate other consequences of an Allee threshold in an evolutionary setting. These include the existence of multiple evolutionarily stable, non-extinction equilibria, and the possibility of evolving to a non-evolutionary stable strategy (ESS) trait from an initial trait near an ESS.     

Multilocus genetics and the coevolution of quantitative traits

We develop and analyze an explicit multilocus genetic model of coevolution. We assume that interactions between two species (mutualists, competitors, or victim and exploiter) are mediated by a pair of additive quantitative traits that are also subject to direct stabilizing selection toward intermediate optima. Using a weak-selection approximation, we derive analytical results for a symmetric case with equal locus effects and no mutation, and we complement these results by numerical simulations of more general cases. We show that mutualistic and competitive interactions always result in coevolution toward a stable equilibrium with no more than one polymorphic locus per species. Victim-exploiter interactions can lead to different dynamic regimes including evolution toward stable equilibria, cycles, and chaos. At equilibrium, the victim is often characterized by a very large genetic variance, whereas the exploiter is polymorphic in no more than one locus. Compared to related one-locus or quantitative genetic models, the multilocus model exhibits two major new properties. First, the equilibrium structure is considerably more complex. We derive detailed conditions for the existence and stability of various classes of equilibria and demonstrate the possibility of multiple simultaneously stable states. Second, the genetic variances change dynamically, which in turn significantly affects the dynamics of the mean trait values. In particular, the dynamics tend to be destabilized by an increase in the number of loci.     

Stability of displacement clines arising in allospecies competition

Summary A stability analysis for geographic displacement clines between competing allospecies is presented. The competition model incorporates the effects of annual dispersal and of selective recruitment determined by geographically varying conditions at the breeding sites. It is assumed that a species gains a local competitive advantage wherever it attains sufficient numerical predominance. This assumption is valid when the species crosspair but the hybrids produced are not recruited into the adult population, because the minority species loses proportionately more of its reproductive potential. It is shown that no stable equilibria occur when the competitive balance is independent of location, but that even a slight geographic variation in the competitive balance allows stable equilibria. The greater the length of the combined breeding ranges of the two species, the smaller the geographic shift in competitive balance needed to produce stability.     

Microbes, models and management of the New Zealand grass grub,

[First paragraph]Improved grasslands pose particular problems in pest management. The areas are extensive and the returns per hectare from grassland agriculture are much less than those from intensive cropping or horticulture, but pastures are usually sown to last for a number of years and have a much more stable ecology than in cropping or horticulture. For these reasons, biological controls and plant resistance have long been the preferred options for managing pasture pests. Within this context, the role of diseases in pasture pest population dynamics has received increasing attention, especially their ability to control pest outbreaks. Diseases are common within our major pasture pest species but their role in population regulation is often difficult to define. One of New Zealand’s major pasture pest species, the grass grub Costelytra zealandica, is widespread and often damaging throughout the country. The insect is an endemic species that has adapted to an introduced pasture system, dominated by exotic plant species (Lolium perene/Trifolium repens), where it can reach populations ten-fold higher than in its native habitat. Such high densities favour disease transmission and it is not surprising that a wide array of pathogens have been recorded from this insect (Glare et al., 1993). But the insect can still be highly damaging and can cause total loss of sown species within 3–4 years from sowing in grass grub prone regions. Probably the most important of these diseases, found throughout New Zealand, is amber disease caused by strains of the bacteria Serratia entomophila and S. proteamaculans. This is an unusual disease, controlled by a bacterial plasmid which has only been found in New Zealand bacterial isolates (Jackson et al., 2001).     

On numerical uncertainty in evaluation of pest population size

Obtaining reliable estimates of pest insect species abundance is an essential part of ecological monitoring programs. It is often the case that data available for obtaining such estimates are sparse which in turn makes achieving an accurate evaluation difficult. This is especially true for strongly heterogeneous pest population density distributions. In our paper we discuss the accuracy of a mean density estimate when a certain class of high aggregation density distributions is considered and a standard statistical method is employed to handle sparse sampled data. It will be shown in the paper that conventional conclusions about the accuracy of the pest population size evaluation do not work when the data are sparse and a new approach is required. Namely, if the number of traps is small, an estimate of the mean density becomes a random variable with an error of high magnitude and we have to compute the probability of an accurate estimate rather than computing the estimate itself. We have obtained a probability of an accurate estimate based on the assumption that only one trap falls within a sub-domain where the pest population density is different from zero. The probability has been calculated for the one-dimensional and the two-dimensional problem.     

Structural Instability and Emergence of the Biodiversity

In the framework of population dynamics, we start from the logistic equation describing the evolution of one species with limited food supply. A split device allows us to consider the population as two sub-populations x and y evolving analogously. The dynamical system has a one-parameter family of equilibria which is structurally unstable. Then small perturbations of the system (describing functional or ethological differentiations between the sub-species) lead in general to a new system involving a fast and a slow dynamics with a finite number of equilibria. In simple situations where the differentiation is clearly either an advantage or an inconvenience for one of the subspecies, the stable equilibrium amounts to extinction of the disadvantaged subspecies (elementary Darwinism). Oppositely more complex differentiations (involving both advantages and inconveniences) often lead to stable equilibria with well-defined non zero proportions of the sub-populations (preservation of the biodiversity). Other examples are concerned with symbiosis-like differentiations, leading to preservation, whereas the opposite case (mutual nuisances) has an unstable equilibrium and lead to extinction of one or the other subspecies according to the initial conditions. The case of a scission into three subspecies is more rich in consequences. In certain cases, predator-prey relations lead to auto-organization phenomena with stable diversity-preserving diversity. Cases of instability are also possible, leading to orbits tending towards a poly-cycle.This implies some kind of pseudo-extinction: this amounts to “pseudo-periodic-like” orbits with “pseudo-periods” larger and larger, tending to infinity; each pseudo-period contains parts where one of the sub-populations practically vanish. Other non-linear perturbations lead to stable orbits.     

Population structure of pierid butterflies

Summary Mark-release-recapture techniques were used to study alfalfa pest populations of Colias philodice eriphyle. Two new methods for estimating relative catchability and residence rates were used to compare males to females. The results show that: (1) both sexes had limited dispersal, with mean individual ranges less than 100 m; (2) males were more abundant than females; (3) males and females had similar residence times; (4) males were more catchable than females in uncut fields, but not in cut fields. Explanations for the differences between the sexes are considered. Females may be less catchable in uncut fields because they spend less time in flight than males. Males may be more abundant than females because they develop faster, and may have lower pre-adult mortality. The differences between the sexes are discussed with respect to reproductive strategy. Comparisons with non-pest C.p. eriphyle show differences between pest and non-pest populations. Pest C.p. eriphyle were more sedentary. The residence times were similar for pest and non-pest populations, but pest C.p. eriphyle probably have longer reproductive life-spans. Mid-summer broods of the pest population were partially overlapping; the non-pest population has discrete broods. Pest population density varied less between years than non-pest population density. The differences between pest and non-pest C.p. eriphyle support the idea of ecological diversity among conspecific populations.     

Pathogen evolution in switching environments: A hybrid dynamical system approach

We propose a hybrid dynamical system approach to model the evolution of a pathogen that experiences different selective pressures according to a stochastic process. In every environment, the evolution of the pathogen is described by a version of the Fisher-Haldane-Wright equation while the switching between environments follows a Markov jump process. We investigate how the qualitative behavior of a simple single-host deterministic system changes when the stochastic switching process is added. In particular, we study the stability in probability of monomorphic equilibria. We prove that in a "constantly" fluctuating environment, the genotype with the highest mean fitness is asymptotically stable in probability while all others are unstable in probability. However, if the probability of host switching depends on the genotype composition of the population, polymorphism can be stably maintained.     

具有经济临界值的非光滑捕食与被捕食系统的全局稳定性分析(英文)

只要害虫种群数量在经济临界值水平之上就连续的实施综合控制策略,基于此本文提出了具有经济临界值的非光滑捕食与被捕食系统.我们给出了系统真平衡态、假平衡态和伪平衡态的存在性和稳定性,以及这些平衡态全局稳定或系统存在全局吸引子的条件,同时借助数值方法验证了所得结论.得到的主要结果说明通过采用临界控制策略能让害虫稳定在一个给定的临界值水平上,而达到害虫控制的目的.     

具有三个年龄阶段的单种群自食模型

建立并研究了两个具有三个年龄阶段的单种群自食模型．这篇文章的主要目的是研究时滞对种群生长的作用,对于没有时滞的的模型,我们利用Liapunov函数,得到了系统平衡点全局渐近稳定的充分条件;而具有时滞的的模型,我们得到,随着时滞T增加,当系数满足一定条件时,正平衡点的稳定性可以改变有限次,最后变成不稳定;否则,时滞模型的正平衡点的稳定性不改变。     

Sex-specific meiotic drive and selection at an imprinted locus

We present a one-locus model that breaks two symmetries of Mendelian genetics. Whereas symmetry of transmission is breached by allowing sex-specific segregation distortion, symmetry of expression is breached by allowing genomic imprinting. Simple conditions for the existence of at least one polymorphic stable equilibrium are provided. In general, population mean fitness is not maximized at polymorphic equilibria. However, mean fitness at a polymorphic equilibrium with segregation distortion may be higher than mean fitness at the corresponding equilibrium with Mendelian segregation if one (or both) of the heterozygote classes has higher fitness than both homozygote classes. In this case, mean fitness is maximized by complete, but opposite, drive in the two sexes. We undertook an extensive numerical analysis of the parameter space, finding, for the first time in this class of models, parameter sets yielding two stable polymorphic equilibria. Multiple equilibria exist both with and without genomic imprinting, although they occurred in a greater proportion of parameter sets with genomic imprinting.     

Estimating Mean First Passage Time of Biased Random Walks with Short Relaxation Time on Complex Networks

Biased random walk has been studied extensively over the past decade especially in the transport and communication networks communities. The mean first passage time (MFPT) of a biased random walk is an important performance indicator in those domains. While the fundamental matrix approach gives precise solution to MFPT, the computation is expensive and the solution lacks interpretability. Other approaches based on the Mean Field Theory relate MFPT to the node degree alone. However, nodes with the same degree may have very different local weight distribution, which may result in vastly different MFPT. We derive an approximate bound to the MFPT of biased random walk with short relaxation time on complex network where the biases are controlled by arbitrarily assigned node weights. We show that the MFPT of a node in this general case is closely related to not only its node degree, but also its local weight distribution. The MFPTs obtained from computer simulations also agree with the new theoretical analysis. Our result enables fast estimation of MFPT, which is useful especially to differentiate between nodes that have very different local node weight distribution even though they share the same node degrees.     

Evolutionary dynamics and strong Allee effects

We describe the dynamics of an evolutionary model for a population subject to a strong Allee effect. The model assumes that the carrying capacity k(u), inherent growth rate r(u), and Allee threshold a(u) are functions of a mean phenotypic trait u subject to evolution. The model is a plane autonomous system that describes the coupled population and mean trait dynamics. We show bounded orbits equilibrate and that the Allee basin shrinks (and can even disappear) as a result of evolution. We also show that stable non-extinction equilibria occur at the local maxima of k(u) and that stable extinction equilibria occur at local minima of r(u). We give examples that illustrate these results and demonstrate other consequences of an Allee threshold in an evolutionary setting. These include the existence of multiple evolutionarily stable, non-extinction equilibria, and the possibility of evolving to a non-evolutionary stable strategy (ESS) trait from an initial trait near an ESS.     

Equilibrium properties of a multi-locus, haploid-selection, symmetric-viability model

Under haploid selection, a multi-locus, diallelic, two-niche Levene (1953) model is studied. Viability coefficients with symmetrically opposing directional selection in each niche are assumed, and with a further simplification that the most and least favored haplotype in each niche shares no alleles in common, and that the selection coefficients monotonically increase or decrease with the number of alleles shared. This model always admits a fully polymorphic symmetric equilibrium, which may or may not be stable.We show that a stable symmetric equilibrium can become unstable via either a supercritical or subcritical pitchfork bifurcation. In the supercritical bifurcation, the symmetric equilibrium bifurcates to a pair of stable fully polymorphic asymmetric equilibria; in the subcritical bifurcation, the symmetric equilibrium bifurcates to a pair of unstable fully polymorphic asymmetric equilibria, which then connect to either another pair of stable fully polymorphic asymmetric equilibria through saddle-node bifurcations, or to a pair of monomorphic equilibria through transcritical bifurcations. As many as three fully polymorphic stable equilibria can coexist, and jump bifurcations can occur between these equilibria when model parameters are varied.In our Levene model, increasing recombination can act to either increase or decrease the genetic diversity of a population. By generating more hybrid offspring from the mating of purebreds, recombination can act to increase genetic diversity provided the symmetric equilibrium remains stable. But by destabilizing the symmetric equilibrium, recombination can ultimately act to decrease genetic diversity.