The exclusion problem in seasonally forced epidemiological systems.

The pathogen exclusion problem is the problem of finding control measures that will exclude a pathogen from an ecological system or, if the system is already disease-free, maintain it in that state. To solve this problem we work within a holistic control theory framework which is consistent with conventional theory for simple systems (where there is no external forcing and constant controls) and seamlessly generalises to complex systems that are subject to multiple component seasonal forcing and targeted variable controls. We develop, customise and integrate a range of numerical and algebraic procedures that provide a coherent methodology powerful enough to solve the exclusion problem in the general case. An important aspect of our solution procedure is its two-stage structure which reveals the epidemiological consequences of the controls used for exclusion. This information augments technical and economic considerations in the design of an acceptable exclusion strategy. Our methodology is used in two examples to show how time-varying controls can exploit the interference and reinforcement created by the external and internal lag structure and encourage the system to 'take over' some of the exclusion effort. On-off control switching, resonant amplification, optimality and controllability are important issues that emerge in the discussion.

Threshold dynamics for periodically forced ecological systems: the control of population invasion and exclusion.

Ecosystems are under increasing threat as a result of anthropogenic activity, through pollution, unregulated harvesting, habitat destruction and the inadvertent spread of pathogens and vertebrate and non-vertebrate species through global transportation links. Many of the necessary interventions to restore or restructure natural ecosystems require the exclusion of a population from the ecosystem or the inclusion of a population if robust biodiversity is the objective. The problem of how best to bring this about is not easy to solve in highly nonlinear systems, especially if the system is exposed to significant time varying external forces. We wish here to build on the understanding gained from previous work by developing an algebraic methodology that yields explicit formulae to analyse the effect of moderate multi-component forcing on the invasion/exclusion process. This can be of assistance to management in designing suitable intervention strategies if one or more of the forcing components is under management control. We apply this methodology to look at three important issues, involving the relationships between resonance and control, between vaccination policy and the stage structure of a disease and between apparent competition and coexistence.

Phase control of resonant systems: interference, chaos and high periodicity.

Much progress has been made in understanding the effect of periodic forcing on epidemiological and ecological systems when that forcing acts on just one part of the system. Much less is known about situations in which several parts of the system are affected. In this case the interaction between the impacts of the different forcing components can lead to reinforcement of system responses or to their interference. This interference phenomenon is significant if some forcing components are anthropogenic for then management might be able to exercise sufficient control to bring about suppression of undesirable aspects of the forcing, for example resonant amplification and the problems this can cause. We set out the algebraic theory when forcing is weak and illustrate by example what can happen when forcing is strong enough to create subharmonics and chaotic states. Phase is the key control variable that can bring about interference, advantageously shift nonlinear response curves and create periodic states out of chaos. The phenomenon in which high period fluctuations appear to be generated by low period forcing is examined and different mechanisms compared in a two-strain epidemiological model. The effect of noise as a source of high period fluctuations is also considered.

Pathogen exclusion from eco-epidemiological systems.

Increasing concerns about the changing environment and the emergence of pathogens that cross species boundaries have added to the urgency of understanding the dynamics of complex ecological systems infected by pathogens. Of particular interest is the often counterintuitive way in which infection and predation interact and the consequent difficulties in designing control strategies to manage the system. To understand the mechanisms involved, we focus on the pathogen exclusion problem, using control maps (on which the network of exclusion thresholds are plotted) in order to readily identify which exclusion strategies will work and why others will not. We apply this approach to the analysis of parasite exclusion in two game bird ecologies. For higher dimensions, we propose a computational scheme that will generate the optimal exclusion strategy, taking into account all operational constraints on the pathogen invasion matrix, populations, and controls. The situation is further complicated when external forcing distorts pathogen thresholds. This distortion is highly sensitive to the lags between forcing components, a sensitivity that can be exploited by management using correctly lagged cyclically varying controls to reduce the effort involved in pathogen exclusion.

Exclusion of generalist pathogens in multihost communities.

Knowing how to control a pathogen that infects more than one host species is of increasing importance because the incidence of such infections grows with continuing environmental change. Of concern are infections transmitted from wildlife to humans or livestock. To determine which options are available to control a pathogen in these circumstances, we analyze the pathogen invasion matrix for the multihost susceptible-infected-susceptible model. We highlight the importance of both community structure and the column sum or row sum index, an indicator of both force of infection and community stability. We derive a set of guidelines for constructing culling strategies and suggest a hybrid strategy that has the advantages of both the bottom-up and the top-down approaches, which we study in some detail. The analysis holds for an arbitrary number of host species, enabling the analysis of large-scale ecological systems and systems with spatial dimensions. We test the robustness of our methods by making two changes in the structure of the underlying dynamic model, adding direct competition and introducing frequency-dependent infection transmission. In particular, we show that the introduction of an additional host can eliminate the pathogen rather than eliminate the resident host. The discussion is illustrated with a reference to bovine tuberculosis.

The evolution of oscillatory behavior in age-structured species.

A major challenge in ecology is to explain why so many species show oscillatory population dynamics and why the oscillations commonly occur with particular periods. The background environment, through noise or seasonality, is one possible driver of these oscillations, as are the components of the trophic web with which the species interacts. However, the oscillation may also be intrinsic, generated by density-dependent effects on the life history. Models of structured single-species systems indicate that a much broader range of oscillatory behavior than that seen in nature is theoretically possible. We test the hypothesis that it is selection that acts to constrain the range of periods. We analyze a nonlinear single-species matrix model with density dependence affecting reproduction and with trade-offs between reproduction and survival. We show that the evolutionarily stable state is oscillatory and has a period roughly twice the time to maturation, in line with observed patterns of periodicity. The robustness of this result to variations in trade-off function and density dependence is tested.

The role of shared parasites in the exclusion of wildlife hosts: Heterakis gallinarum in the ring-necked pheasant and the grey partridge.

1. A two-host shared-macroparasite model was parameterized from the results of infection and transmission experiments, to investigate whether apparent competition between the ring-necked pheasant (Phasianus colchicus) and the grey partridge (Perdix perdix), mediated via the shared nematode Heterakis gallinarum, could theoretically cause partridge exclusion. 2. Both the model created and the experiments conducted show that the bulk of H. gallinarum infection to partridges, when they occur in the same locations as pheasants, will be from the pheasants and not from the partridges themselves. This is due to R0 for the parasite being 1·23 when infecting pheasants, but only 0·0057 when infecting partridges. Thus, when the pheasant is present in the model the partridge population is impacted by the shared parasite but, when the pheasant is absent, the parasite is lost from the system. 3. Based on best available parameter estimates, the observed impact of H. gallinarum on the grey partridge may be sufficient to cause exclusion when the pheasant is present in the model. This supports the hypothesis that the UK grey partridge decline observed over the past 50 years may be partly due to apparent competition with pheasants. 4. Habitat separation between the two host species, where it decreases the rate of H. gallinarum transmission from the pheasant to the partridge, may allow them to co-exist in the field in the presence of the parasite. We predict, however, that grey partridge exclusion would still occur if separation was less than 43%.

Environmental forcing, invasion and control of ecological and epidemiological systems.

Destabilising a biological system through periodic or stochastic forcing can lead to significant changes in system behaviour. Forcing can bring about coexistence when previously there was exclusion; it can excite massive system response through resonance, it can offset the negative effect of apparent competition and it can change the conditions under which the system can be invaded. Our main focus is on the invasion properties of continuous time models under periodic forcing. We show that invasion is highly sensitive to the strength, period, phase, shape and configuration of the forcing components. This complexity can be of great advantage if some of the forcing components are anthropogenic in origin. They can be turned into instruments of control to achieve specific objectives in ecology and disease management, for example. Culling, vaccination and resource regulation are considered. A general analysis is presented, based on the leading Lyapunov exponent criterion for invasion. For unstructured invaders, a formula for this exponent can typically be written down from the model equations. Whether forcing hinders or encourages invasion depends on two factors: the covariances between invader parameters and resident populations and the shifts in average resident population levels brought about by the forcing. The invasion dynamics of a structured invader are much more complicated but an analytic solution can be obtained in quadratic approximation for moderate forcing strength. The general theory is illustrated by a range of models drawn from ecology and epidemiology. The relationship between periodic and stochastic forcing is also considered.

The impact of environmental fluctuations on structured discrete time population models: resonance, synchrony and threshold behaviour.

External forcing of a discrete time ecological system does not just add variation to existing dynamics but can change the dynamics. We study the mechanisms that can bring this about, focusing on the key concepts of excitation and suppression which emerge when analysing the power spectra of the system in linear approximation. Excitation, through resonance between the system dynamics and the external forcing, is the greater the closer the system is to the boundary of the stability region. This amplification means that the extinction of populations becomes possible sooner than expected and, conversely, invasion can be significantly delayed. Suppression and the consequent redistribution of power within the spectrum proves to be a function both of the connectivity of the network graph of the system and the way that external forcing is applied to the system. It is also established that colour in stochastic forcing can have a major impact, by enhancing resonance and by greater redistribution of power. This can mean a higher risk of extinction through larger fluctuations in population numbers and a higher degree of synchrony between populations. The implications of external forcing for stage-structured species, for populations in competition and for trophic web systems are studied using the tools and concepts developed in the paper.

The amplification of environmental noise in population models: causes and consequences.

Environmental variability is a ubiquitous feature of every organism's habitat. However, the interaction between density dependence and those density-independent factors that are manifested as environmental noise is poorly understood. We are interested in the conditions under which noise interacts with the density dependence to cause amplification of that noise when filtered by the system. For a broad family of structured population models, we show that amplification occurs near the threshold from stable to unstable dynamics by deriving an analytic formula for the amplification under weak noise. We confirm that the effect of noise is to sustain oscillations that would otherwise decay, and we show that it is the amplitude and not the phase that is affected. This is a feature noted in several recent studies. We study this phenomenon in detail for the lurchin and LPA models of population dynamics. We find that the degree of amplification is sensitive to both the noise input and life-history stage through which it acts, that the results hold for surprisingly high levels of noise, and that stochastic chaos (as measured by local Lyapunov exponents) is a concomitant feature of amplification. Further, it is shown that the temporal autocorrelation, or "color," of the noise has a major impact on the system response. We discuss the conditions under which color increases population variance and hence the risk of extinction, and we show that periodicity is sharpened when the color of the noise and dynamics coincide. Otherwise, there is interference, which shows how difficult it is in practice to separate the effects of nonlinearity and noise in short time series. The sensitivity of the population dynamics to noise when close to a bifurcation has wide-ranging consequences for the evolution and ecology of population dynamics.

Large amplification in stage-structured models: Arnol'd tongues revisited.

The coexistence of periodic and point attractors has been confirmed for a range of stage-structured discrete time models. The periodic attractor cycles have large amplitude, with the populations cycling between extremely low and surprisingly high values when compared to the equilibrium level. In this situation a stable state can be shocked by noise of sufficient strength into a state of high volatility. We found that the source of these large amplitude cycles are Arnol'd tongues, special regions of parameter space where the system exhibits periodic behaviour. Most of these tongues lie entirely in that part of parameter space where the system is unstable, but there are exceptions and these exceptions are the tongues that lead to attractor coexistence. Similarity in the geometry of Arnol'd tongues over the range of models considered might suggest that this is a common feature of stage-structured models but in the absence of proof this can only be a useful working hypothesis. The analysis shows that although large amplitude cycles might exist mathematically they might not be accessible biologically if biological constraints, such as non-negativity of population densities and vital rates, are imposed. Accessibility is found to be highly sensitive to model structure even though the mathematical structure is not. This highlights the danger of drawing biological conclusions from particular models. Having a comprehensive view of the different mechanisms by which periodic states can arise in families of discrete time models is important in the debate on whether the causes of periodicity in particular ecological systems are intrinsic, environmental or trophic. This paper is a contribution to that continuing debate.

Host exclusion and coexistence in apparent and direct competition: An application of bifurcation theory.

Recent empirical studies have focused attention on the interplay in multi-host systems of parasite-mediated apparent competition and direct competition between hosts. However, theoretical investigation of such systems has been hindered by the onset of algebraic intractability with the increase in system dimensionality. In this paper we circumvent this problem by using a geometric approach in which arrays of bifurcation maps are constructed, each map being structured by the set of (bifurcation) points in parameter space at which qualitative changes in system behaviour take place. From these maps can be compiled a concise catalogue of the possible modes of system behaviour, enabling an investigation of the interaction of apparent and direct competitive forces to be carried out. Of importance is the identification of those situations where increasing one or both of these competitive forces leads to a change in the stability state. The maps provide an efficient way of determining whether, and, if so, under what conditions, specific modes of behaviour are allowed by the model. Two field phenomena of particular interest, discussed in the paper, are host invasion and dominance reversal resulting from the introduction of the pathogen into a directly competitive system.

Parasite-mediated and direct competition in a two-host shared macroparasite system.

This paper investigates the local dynamical behaviour of a deterministic model describing two host species experiencing three forms of competition: direct competition, apparent competition mediated by macroparasites, and intra-specific (density-dependent) competition. The problem of algebraic intractability is sidestepped by adopting a geometric approach, in which an array of maps is constructed in parameter space, each structured by bifurcation surfaces which mark qualitative changes in system behaviour. The maps provide both a succinct and a comprehensive overview of the stability and feasibility structure of the system equilibria, from which can be deduced the possible modes of local dynamical behaviour. A detailed examination of these maps shows that (i) the system is highly sensitive to the effect of infection on fecundity with synchronous sustained cycles readily generated by Hopf bifurcations; (ii) for a broad range of parameter values, pertinent to actual biological systems, apparent competition mediated by macroparasites is sufficient, on its own, to explain host exclusion; (iii) direct competition reinforces parasite-mediated competition to expand the host exclusion region; and (iv) the condition for host exclusion can be expressed simply in a form which holds for both micro- and macroparasite models and which involves just two key indices, measuring tolerance to the infection and the strength of direct competition. The techniques used in this paper are not restricted to the analysis of host-parasite systems but can be applied to a wide range of nonlinear population models. They are therefore as relevant to the analysis of such general issues as exploitative competition and trophic interactions as they are to specific epidemiological problems.

Multihost, multiparasite systems: an application of bifurcation theory.

The local analysis of multihost multiparasite models has been hampered by algebraic intractability. There have been two responses to this difficulty: extensive numerical investigation, and simplification to a level where analytical techniques work. In this paper we describe another approach, based on bifurcation theory, in which the qualitative properties of the model equilibrium structure are realized on an array of maps drawn in parameter space. This approach is described in the context of two models: the basic two-host shared microparasite S-I model and the single-host two-microparasite S-I (susceptible-infective) model. The procedure involved does not require model simplification through a reduction in dimensionality. It can handle intraspecific as well as parasite-mediated competition and, in the second model, single-host parasite coexistence. The map arrays provide a concise catalogue of the possible modes of behaviour of a system and an explanation for changes in that behaviour. In particular, the reasons why the conjectures made about the behaviour of the first of these models do not hold throughout parameter space are immediately clear from the map structure, as are the conditions for collusive and competitive behaviour between the two types of parasite in the second model.

The effect of seasonal host birth rates on population dynamics: the importance of resonance.

Many of the simple mathematical models currently in use often fail to capture important biological factors. Here we extend current models of insect-pathogen interactions to include seasonality in the birth rate. In particular, we consider the SIR model with self-regulation when applied to specific cases--rabbit haemorrhagic disease and fox rabies. In this paper, we briefly summarize the results of the model with a constant time-independent birth rate, a, which we then replace with the time dependent birth rate a(t), to investigate how this effects the dynamics of the host population. We can split parameter space into an area in which the model without seasonality has no oscillations, in which case a simple averaging rule predicts the behaviour. Alternatively, in the area where oscillations to the equilibrium do occur in the non-seasonal model, disease persistence is more complicated and we get more complex dynamical behaviour in this case. We apply resonance techniques to discover the structure of the subharmonic modes of the SIR model with self-regulation. We then look at whether many biological systems are likely to display these "resonant" dynamics and find that we would expect them to be widespread.

Differential impact of a shared nematode parasite on two gamebird hosts: implications for apparent competition.

If the deleterious effects of non-specific parasites are greater on vulnerable host species than on reservoir host species then exclusion of the vulnerable host through apparent competition is more likely. Evidence suggests that such a mechanism occurs in interactions between the ring-necked pheasant (Phasianus colchicus), the grey partridge (Perdix perdix), and their shared caecal nematode Heterakis gallinarum. Modelling of the system predicts that the reduced parasite impact on the pheasant compared to the partridge results in the force of infection transmitted from pheasants to partridges being sufficient to cause partridge exclusion. Since the parasite impacts are currently estimated from correlational work, controlled infections were conducted to experimentally compare the impact of H. gallinarum on the two hosts and verify cause and effect. While challenged partridges showed reduced mass gain, decreased food consumption, and impaired caecal activity, in comparison to controls, the only detectable effect of parasite challenge on the pheasant was impaired caecal activity. The impact of H. gallinarum on challenged partridges conforms with previous correlational data, supporting the prediction that parasite-mediated apparent competition with the ring-necked pheasant may result in grey partridge exclusion. However, the observed decrease in the caecal activity of challenged pheasants could imply that H. gallinarum may also have an impact on the fecundity and survival of pheasants in the wild, particularly if food is limiting. If this is the case, the associated decrease in the force of infection to which the partridge is exposed may be sufficient to change the model prediction from partridge exclusion to pheasant and partridge coexistence.