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Phenotypic plasticity plays an important role in the survival of individuals. In microbial host-virus systems, previous studies have shown the stabilizing effect that host plasticity has on the coexistence of the system. By contrast, it remains uncertain how the dependence of the virus on the metabolism of the host (i.e. “viral plasticity”) shapes bacteria-phage population dynamics in general, or the stability of the system in particular. Moreover, bacteria-phage models that do not consider viral plasticity are now recognised as overly simplistic. For these reasons, here we focus on the effect of viral plasticity on the stability of the system under different environmental conditions. We compared the predictions from a standard bacteria-phage model, which neglects plasticity, with those of a modification that includes viral plasticity. We investigated under which conditions viral plasticity promotes coexistence, with or without oscillatory dynamics. Our analysis shows that including viral plasticity reveals coexistence in regions of the parameter space where models without plasticity predict a collapse of the system. We also show that viral plasticity tends to reduce population oscillations, although this stabilizing effect is not consistently observed across environmental conditions: plasticity may instead reinforce dynamic feedbacks between the host, the virus, and the environment, which leads to wider oscillations. Our results contribute to a deeper understanding of the dynamic control of bacteriophage on host populations observed in nature.

Understanding the persistence and growth of natural populations in environments subject to random localised change is relevant both to the conservation of threatened species and to the control of invasive species. By developing and analysing simple strategic growth models in environments subject to random fragmentation events, we show that simple approximations can be used to predict invasion speeds and extinction probabilities. The rate and size of fragmentation events interact in a nonlinear way, a finding with important consequences for the efficient control of invasive species. Infrequent, large-scale fragmentation events provide more effective means of control than more frequent, smaller scale efforts.

With the recent resurgence of vector-borne diseases due to urbanization and development there is an urgent need to understand the dynamics of vector-borne diseases in rapidly changing urban environments. For example, many empirical studies have produced the disturbing finding that diseases continue to persist in modern city centers with zero or low rates of transmission. We develop spatial models of vector-borne disease dynamics on a network of patches to examine how the movement of humans in heterogeneous environments affects transmission. We show that the movement of humans between patches is sufficient to maintain disease persistence in patches with zero transmission. We construct two classes of models using different approaches: (i) Lagrangian models that mimic human commuting behavior and (ii) Eulerian models that mimic human migration. We determine the basic reproduction number R(0) for both modeling approaches. We show that for both approaches that if the disease-free equilibrium is stable (R(0)1) then there exists a unique positive (endemic) equilibrium that is globally stable among positive solutions. Finally, we prove in general that Lagrangian and Eulerian modeling approaches are not equivalent. The modeling approaches presented provide a framework to explore spatial vector-borne disease dynamics and control in heterogeneous environments. As an example, we consider two patches in which the disease dies out in both patches when there is no movement between them. Numerical simulations demonstrate that the disease becomes endemic in both patches when humans move between the two patches. National Institutes of Health/[P20-RR020770]/NIH/Estados Unidos National Science Foundation/[DMS-0514839]/NSF/Estados Unidos National Science Foundation/[DMS-0816068]/NSF/Estados Unidos National Science Foundation/[DMS-0715772]/NSF/Estados Unidos UCR::Vicerrectoría de Investigación::Unidades de Investigación::Ciencias de la Salud::Centro de Investigación en Enfermedades Tropicales (CIET)

Two definitions of persistence despite perturbations in deterministic models are presented. The first definition, persistence despite frequent small perturbations, is shown to be equivalent to the existence of a positive attractor i.e. an attractor bounded away from extinction. The second definition, persistence despite rare large perturbations, is shown to be equivalent to permanence i.e. a positive attractor whose basin of attraction includes all positive states. Both definitions set up a natural dichotomy for classifying models of interacting populations. Namely, a model is either persistent despite perturbations or not. When it is not persistent, it follows that all initial conditions are prone to extinction due to perturbations of the appropriate type. For frequent small perturbations, this method of classification is shown to be generically robust: there is a dense set of models for which persistent (respectively, extinction prone) models lies within an open set of persistent (resp. extinction prone) models. For rare large perturbations, this method of classification is shown not to be generically robust. Namely, work of Josef Hofbauer and the author have shown there are open sets of ecological models containing a dense sets of permanent models and a dense set of extinction prone models. The merits and drawbacks of these different definitions are discussed.

This paper analyses the effect of migration frequency on the stability and persistence of a host-parasitoid system in a two-patch environment. The hosts and parasitoids are allowed to move from one patch to the other a certain number of times within a generation. When this number is low, i.e. when the time-scales associated with migration and demography are of the same order, host-parasitoid interactions are usually not persistent. When this number is high, however, persistence is more likely. Moreover, in this situation, aggregation methods can be used to simplify the proposed initial model into an aggregated model describing the dynamics of both the total host and parasitoid populations. Analysis of the aggregated model shows that the system reaches a stable steady state for some regions of the parameter domain. Persistence occurs when the movement of the parasitoids is asymmetrical, i.e. they move preferentially to one of the two patches. We show that the growth rate of the host population is a key parameter in determining which migration strategies of the parasitoids lead to persistent host-parasitoid interactions.

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.

The dynamical theory of food webs has been based typically on local stability analysis. The relevance of local stability to food web properties has been questioned because local stability holds only in the immediate vicinity of the equilibrium and provides no information about the size of the basin of attraction. Local stability does not guarantee persistence of food webs in stochastic environments. Moreover, local stability excludes more complex dynamics such as periodic and chaotic behaviors, which may allow persistence. Global stability and permanence could be better criteria of community persistence. Our simulation analysis suggests that these three stability measures are qualitatively consistent in that all three predict decreasing stability with increasing complexity. Some new predictions on how stability depends on food web configurations are generated here: a consumer–victim link has a smaller effect on the probabilities of stability, as measured by all three stability criteria, than a pair of recipient-controlled and donor-controlled links; a recipient-controlled link has a larger effect on the probabilities of local stability and permanence than a donor-controlled link, while they have the same effect on the probability of global stability; food webs with equal proportions of donor-controlled and recipient-controlled links are less stable than those with different proportions.

Habitat fragmentation is generally considered to be detrimental to the persistence of natural populations. In nature management, one therefore tends to prefer few large nature reserves over many small nature reserves having equal total area. This paper examines whether this preference is warranted in a metapopulation framework with circular reserves (patches) by formulating the dependence of metapopulation persistence on the size and number of reserves, both of which depend on reserve radius if the total area is kept constant. Two measures of metapopulation persistence are used: R(0), the number of patches colonized by an occupied patch during its lifetime as an occupied patch, and T(e), the expected time to extinction. These two measures are functions of the extinction and colonization rates of the metapopulation. Several mechanisms for the extinction and colonization processes are formulated from which the dependence of these rates on reserve radius is calculated. It turns out that T(e)generally increases with reserve radius for all mechanisms, which supports the preference of few large reserves. However, R(0)supports this preference only in the case of some special, rather unrealistic, mechanisms. In many other, more realistic, cases an intermediate reserve size exists for which metapopulation persistence measured by R(0)is optimal.

Recently, there has been a great deal of interest in the dynamics of metapopulations,where a number of local populations are coupled via dispersal. The importance of movement for the persistence of an ensemble of locally unstable patches has been established in many studies. In this paper, we present analytical and simulation results concerning the effects of spatial structure on the equilibrium stability of individual populations. We conclude that for general single-species and two-species competition models, the introduction of the spatial dimension in a biologically sensible way has no effect on the overall stability properties. In host-parasitoid models, however, strong host or parasitoid over-dispersal may be destabilising.

A simple population model is presented that exhibits both the period-doubling and the quasi-periodic routes to chaos with the variation of a single parameter. This fact is exploited to contrast the relative robustness of the two routes against structural perturbations in the form of added immigration terms. It is found that although sufficiently large levels of immigration eventually suppress chaotic dynamics in this model, chaos reached via the period-doubling route is much more easily inhibited than via the quasi-periodic one.

Metapopulation theory for a long time has assumed dispersal to be symmetric, i.e. patches are connected through migrants dispersing bi-directionally without a preferred direction. However, for natural populations symmetry is often broken, e.g. for species in the marine environment dispersing through the transport of pelagic larvae with ocean currents. The few recent studies of asymmetric dispersal concluded, that asymmetry has a distinct negative impact on the persistence of metapopulations. Detailed analysis however revealed, that these previous studies might have been unable to properly disentangle the effect of symmetry from other potentially confounding properties of dispersal patterns. We resolve this issue by systematically investigating the symmetry of dispersal patterns and its impact on metapopulation persistence. Our main analysis based on a metapopulation model equivalent to previous studies but now applied on regular dispersal patterns aims to isolate the effect of dispersal symmetry on metapopulation persistence. Our results suggest, that asymmetry in itself does not imply negative effects on metapopulation persistence. For this reason we recommend to investigate it in connection with other properties of dispersal instead of in isolation., 19 pages, 5 figures

There has been great interest in the invasion and persistence of algal and insect populations in rivers. Recent modeling approaches assume that the flow speed of the river is constant. In reality, however, flow speeds in rivers change significantly on various temporal scales due to seasonality, weather conditions, or many human activities such as hydroelectric dams. In this paper, we study persistence conditions by deriving the upstream invasion speed in simple reaction–advection–diffusion equations with coefficients chosen to be periodic step functions. The key methodological idea to determine the spreading speed is to use the exponential transform in order to obtain a moment generating function. In a temporally periodic environment, the averages of each coefficient function determine the minimal upstream and downstream propagation speeds for a single-compartment model. For a two-compartment model, the temporal variation can enhance population persistence.

This is a continuation of our paper [Liu, M., Wang, K., 2010. Persistence and extinction of a stochastic single-species model under regime switching in a polluted environment, J. Theor. Biol. 264, 934-944]. Taking both white noise and colored noise into account, a stochastic single-species model under regime switching in a polluted environment is studied. Sufficient conditions for extinction, stochastic nonpersistence in the mean, stochastic weak persistence and stochastic permanence are established. The threshold between stochastic weak persistence and extinction is obtained. The results show that a different type of noise has a different effect on the survival results.

The assumption that populations are completely mixed is reasonable for many populations, but there is likely to be some degree of local interaction whether spatially or socially in many systems. An important question is therefore how strong these local interactions need to be before there are significant effects on the dynamics of the system. Here, our approach is to use a multi-scale pair-approximation model to move between completely local and completely mixed host-parasite interactions. We show that systems dominated by near neighbour effects have less persistence of disease, and a greater possibility of parasite driven extinction and limit cycles. Furthermore this reduction in persistence occurs over a wide range of infection scales and is still significant in predominantly mixed host populations. Deterministic extinctions are only likely in highly spatial SI systems while oscillations also persist over a wide range of infection ranges, but only in hosts that reproduce mostly locally. In general the mean-field may well be a good approximation for many systems, even when there are a significant proportion of near neighbour events, but this depends crucially on the ecological context.

Spatial heterogeneity is believed to play an important role in the persistence and dynamics of epidemics of childhood diseases because asynchrony between populations within different regions allows global persistence, even if the disease dies out locally. A simple multi-patch (metapopulation) model for spatial heterogeneity in epidemics is analysed and we examine conditions under which patches become synchronised. We show that the patches in non-seasonal deterministic models often oscillate in phase for all but the weakest between patch coupling. Synchronisation is also seen for stochastic models, although slightly stronger coupling is needed to overcome the random effects. We demonstrate that the inclusion of seasonal forcing in deterministic models can lead to the maintenance of phase differences between patches. Complex dynamic behaviour is observed in the seasonally forced spatial model, along with the coexistence of many different behaviours. Compared to the non-spatial model, chaotic solutions are observed for weaker seasonal forcing; these solutions have a more realistic minimum number of infectives.

An attempt is made to formulate the total tunnel existing after a given time in flour containing the beetle Tribolium confusum . Tunnels in the flour mass are destroyed by transversal of the existing tunnels during boring of new ones. The half lives of tunnels are evaluated and solutions are sought in exponential equations describing the fate of various types of tunnel after a given interval. A numerical example is given.

I have constructed a spatially distributed analytical model of predators, superior prey competitors, and inferior prey competitors, based on the limiting deterministic version of a simulation model by Caswell (1978). Persistence regions for the three populations are mapped in parameter space. Conceptually shrinking the system from infinite size (i.e. infinitely many spatial “cells”) to some finite size introduces demographic stochasticity, increasing the chance of extinction of one or more populations within a given time interval. But some of the finite (stochastic) system's behavior, such as any tendency to damp perturbations, can be related to the behavior of the deterministic system at the same location in parameter space.