Migration frequency and the persistence of host-parasitoid interactions

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.     

Host-Parasitoid Dynamics and the Success of Biological Control When Parasitoids Are Prone to Allee Effects

In sexual organisms, low population density can result in mating failures and subsequently yields a low population growth rate and high chance of extinction. For species that are in tight interaction, as in host-parasitoid systems, population dynamics are primarily constrained by demographic interdependences, so that mating failures may have much more intricate consequences. Our main objective is to study the demographic consequences of parasitoid mating failures at low density and its consequences on the success of biological control. For this, we developed a deterministic host-parasitoid model with a mate-finding Allee effect, allowing to tackle interactions between the Allee effect and key determinants of host-parasitoid demography such as the distribution of parasitoid attacks and host competition. Our study shows that parasitoid mating failures at low density result in an extinction threshold and increase the domain of parasitoid deterministic extinction. When proned to mate finding difficulties, parasitoids with cyclic dynamics or low searching efficiency go extinct; parasitoids with high searching efficiency may either persist or go extinct, depending on host intraspecific competition. We show that parasitoids suitable as biocontrol agents for their ability to reduce host populations are particularly likely to suffer from mate-finding Allee effects. This study highlights novel perspectives for understanding of the dynamics observed in natural host-parasitoid systems and improving the success of parasitoid introductions.     

Spatial mechanisms for coexistence of species sharing a common natural enemy

We examine the conditions under which spatial structure can mediate coexistence of apparent competitors. We use a spatially explicit, host-parasitoid metapopulation model incorporating local dynamics of Nicholson-Bailey type and global dispersal. Depending on the model parameters, the resulting system displays a plethora of asynchronous dynamical behaviors for which permanent or transient coexistence is observed. We identify a number of spatially mediated tradeoffs which apparent competitors can utilize and demonstrate that the dynamics of spatial coexistence can typically be understood from consideration of two and three patch systems. The phase relationships of species abundances are different for our model than for some other mechanisms of spatial coexistence. We discuss the implications of our findings relative to issues of community organization and biological conservation.     

Periodicity, persistence, and collapse in host-parasitoid systems with egg limitation

There is an emerging consensus that parasitoids are limited by the number of eggs which they can lay as well as the amount of time they can search for their hosts. Since egg limitation tends to destabilize host-parasitoid dynamics, successful control of insect pests by parasitoids requires additional stabilizing mechanisms such as heterogeneity in the distribution of parasitoid attacks and host density-dependence. To better understand how egg limitation, search limitation, heterogeneity in parasitoid attacks, and host density-dependence influence host-parasitoid dynamics, discrete time models accounting for these factors are analyzed. When parasitoids are purely egg-limited, a complete anaylsis of the host-parasitoid dynamics are possible. The analysis implies that the parasitoid can invade the host system only if the parasitoid's intrinsic fitness exceeds the host's intrinsic fitness. When the parasitoid can invade, there is a critical threshold, CV*>1, of the coefficient of variation (CV) of the distribution of parasitoid attacks that determines that outcome of the invasion. If parasitoid attacks sufficiently aggregated (i.e., CV>CV*), then the host and parasitoid coexist. Typically (in a topological sense), this coexistence is shown to occur about a periodic attractor or a stable equilibrium. If the parasitoid attacks are sufficiently random (i.e. CV1. When CV<1, the parasitoid exhibits highly oscillatory dynamics. Alternatively, when parasitoid attacks are sufficiently aggregated but not overly aggregated (i.e. CV>1 but close to 1), the host and parasitoid coexist about a stable equilibrium with low host densities. The implications of these results for classical biological control are discussed.     

The Role of Spatial Refuges in Coupled Map Lattice Model for Host-Parasitoid Systems

A Coupled Map Lattice (CML) model, for host-parasitoid Nicholson–Bailey interactions, with an explicit spatial distribution of partial refuge areas, is presented by considering the parasitoid attack rate as a patch dependent parameter. The effect of habitat heterogeneity on the dynamics of both populations, that is, on their spatial distribution and temporal behavior is analyzed. Our results show that depending on many features such as position, size, and fragmentation of a refuge, as well as the dispersal parameters of hosts and parasitoids, together with the parasitoid attack rate, the inclusion of refuges may as well stabilize as destabilize the host-parasitoid dynamics. The results are analyzed for the local and the global scales. Spatial patterns resulting from such heterogeneous patchy environments are also obtained.     

Discrete-time host-parasitoid models with pest control

We propose a simple discrete-time host-parasitoid model to investigate the impact of external input of parasitoids upon the host-parasitoid interactions. It is proved that the input of the external parasitoids can eventually eliminate the host population if it is above a threshold and it also decreases the host population level in the unique interior equilibrium. It can simplify the host-parasitoid dynamics when the host population practices contest competition. We then consider a corresponding optimal control problem over a finite time period. We also derive an optimal control model using a chemical as a control for the hosts. Applying the forward-backward sweep method, we solve the optimal control problems numerically and compare the optimal host populations with the host populations when no control is applied. Our study concludes that applying a chemical to eliminate the hosts directly may be a more effective control strategy than using the parasitoids to indirectly suppress the hosts.     

Dynamic heterogeneous spatio-temporal pattern formation in host-parasitoid systems with synchronised generations

In this paper we develop a general mathematical model describing the spatio-temporal dynamics of host-parasitoid systems with forced generational synchronisation, for example seasonally induced diapause. The model itself may be described as an individual-based stochastic model with the individual movement rules derived from an underlying continuum PDE model. This approach permits direct comparison between the discrete model and the continuum model. The model includes both within-generation and between-generation mechanisms for population regulation and focuses on the interactions between immobile juvenile hosts, adult hosts and adult parasitoids in a two-dimensional domain. These interactions are mediated, as they are in many such host-parasitoid systems, by the presence of a volatile semio-chemical (kairomone) emitted by the hosts or the hosts food plant. The model investigates the effects on population dynamics for different host versus parasitoid movement strategies as well as the transient dynamics leading to steady states. Despite some agreement between the individual and continuum models for certain motility parameter ranges, the model dynamics diverge when host and parasitoid motilities are unequal. The individual-based model maintains spatially heterogeneous oscillatory dynamics when the continuum model predicts a homogeneous steady state. We discuss the implications of these results for mechanistic models of phenotype evolution.P. Schofield gratefully acknowledges the financial support of the BBSRC and The Wellcome Trust.     

The role of prey taxis in biological control: a spatial theoretical model

We study a reaction-diffusion-advection model for the dynamics of populations under biological control. A control agent is assumed to be a predator species that has the ability to perceive the heterogeneity of pest distribution. The advection term represents the predator density movement according to a basic prey taxis assumption: acceleration of predators is proportional to the prey density gradient. The prey population reproduces logistically, and the local population interactions follow the Holling Type II trophic function. On the scale of the population, our spatially explicit approach subdivides the predation process into random movement represented by diffusion, directed movement described by prey taxis, local prey encounters, and consumption modeled by the trophic function. Thus, our model allows studying the effects of large-scale predator spatial activity on population dynamics. We show under which conditions spatial patterns are generated by prey taxis and how this affects the predator ability to maintain the pest population below some economic threshold. In particular, intermediate taxis activity can stabilize predator-pest populations at a very low level of pest density, ensuring successful biological control. However, very intensive prey taxis destroys the stability, leading to chaotic dynamics with pronounced outbreaks of pest density.     

Mathematical modelling of host-parasitoid systems: effects of chemically mediated parasitoid foraging strategies on within- and between-generation spatio-temporal dynamics.

In this paper we develop a novel discrete, individual-based mathematical model to investigate the effect of parasitoid foraging strategies on the spatial and temporal dynamics of host-parasitoid systems. The model is used to compare na?ve or random search strategies with search strategies that depend on experience and sensitivity to semiochemicals in the environment. It focuses on simple mechanistic interactions between individual hosts, parasitoids, and an underlying field of a volatile semiochemical (emitted by the hosts during feeding) which acts as a chemoattractant for the parasitoids. The model addresses movement at different spatial scales, where scale of movement also depends on the internal state of an individual. Individual interactions between hosts and parasitoids are modelled at a discrete (micro-scale) level using probabilistic rules. The resulting within-generation dynamics produced by these interactions are then used to generate the population levels for successive generations. The model simulations examine the effect of various key parameters of the model on (i) the spatio-temporal patterns of hosts and parasitoids within generations; (ii) the population levels of the hosts and parasitoids between generations. Key results of the model simulations show that the following model parameters have an important effect on either the development of patchiness within generations or the stability/instability of the population levels between generations: (i) the rate of diffusion of the kairomones; (ii) the specific search strategy adopted by the parasitoids; (iii) the rate of host increase between successive generations. Finally, evolutionary aspects concerning competition between several parasitoid subpopulations adopting different search strategies are also examined.     

Host-parasitoid spatial dynamics in heterogeneous landscapes

This paper explores the effect of spatial processes in a heterogeneous environment on the dynamics of a host-parasitoid interaction. The environment consists of a lattice of favourable (habitat) and hostile (matrix) hexagonal cells, whose spatial distribution is measured by habitat proportion and spatial autocorrelation (inverse of fragmentation). At each time step, a fixed fraction of both populations disperses to the adjacent cells where it reproduces following the Nicholson-Bailey model. Aspects of the dynamics analysed include extinction, stability, cycle period and amplitude, and the spatial patterns emerging from the dynamics.
We find that, depending primarily on the fraction of the host population that disperses in each generation and on the landscape geometry, five classes of spatio-temporal dynamics can be objectively distinguished: spatial chaos, spirals, metapopulation, mainland-island and spiral fragments. The first two are commonly found in theoretical studies of homogeneous landscapes. The other three are direct consequences of the heterogeneity and have strong similarities to dynamic patterns observed in real systems (e.g. extinction-recolonisation, source-sink, outbreaks, spreading waves).
We discuss the processes that generate these patterns and allow the system to persist. The importance of these results is threefold: first, our model merges into a same theoretical framework dynamics commonly observed in the field that are usually modelled independently. Second, these dynamics and patterns are explained by dispersal rate and common landscape statistics, thus linking in a practical way population ecology to landscape ecology. Third, we show that the landscape geometry has a qualitative effect on the length of the cycles and, in particular, we demonstrate how very long periods can be produced by spatial processes.     

Establishment of a rice transgene flow model for predicting maximum distances of gene flow in southern China

We aimed to establish a rice gene flow model based on (i) the Gaussian plume model, (ii) data from a three-location x 3-yr field experiment on transgene flow to common rice cultivars (Oryza sativa), male sterile (ms) lines (O. sativa) and common wild rice (Oryza rufipogon), and (iii) 32-yr historical meteorological data collected from 38 meteorological stations in southern China during the rice flowering period. The concept of the gene flow coefficient (GFC) is proposed; that is, the ratio of the transgene flow frequency (G%) obtained from field experiments to the aggregated pollen dispersal frequency (P%) calculated based on the pollen dispersal model. The maximum distances of gene flow (MDGF) to traditional rice cultivars, ms lines, and common wild rice at a threshold value of either 1.0 or 0.1% were determined. The MDGF and its spatial distribution in southern China show that the gene flow pattern is significantly affected by the monsoon climate, the topography, and the outcrossing ability of recipients. We believe that the information provided in this study will be useful for the risk assessment of transgenic rice in other rice-growing regions.     

Deriving fine-scale models of human mobility from aggregated origin-destination flow data

The spatial dynamics of epidemics are fundamentally affected by patterns of human mobility. Mobile phone call detail records (CDRs) are a rich source of mobility data, and allow semi-mechanistic models of movement to be parameterised even for resource-poor settings. While the gravity model typically reproduces human movement reasonably well at the administrative level spatial scale, past studies suggest that parameter estimates vary with the level of spatial discretisation at which models are fitted. Given that privacy concerns usually preclude public release of very fine-scale movement data, such variation would be problematic for individual-based simulations of epidemic spread parametrised at a fine spatial scale. We therefore present new methods to fit fine-scale mathematical mobility models (here we implement variants of the gravity and radiation models) to spatially aggregated movement data and investigate how model parameter estimates vary with spatial resolution. We use gridded population data at 1km resolution to derive population counts at different spatial scales (down to ∼ 5km grids) and implement mobility models at each scale. Parameters are estimated from administrative-level flow data between overnight locations in Kenya and Namibia derived from CDRs: where the model spatial resolution exceeds that of the mobility data, we compare the flow data between a particular origin and destination with the sum of all model flows between cells that lie within those particular origin and destination administrative units. Clear evidence of over-dispersion supports the use of negative binomial instead of Poisson likelihood for count data with high values. Radiation models use fewer parameters than the gravity model and better predict trips between overnight locations for both considered countries. Results show that estimates for some parameters change between countries and with spatial resolution and highlight how imperfect flow data and spatial population distribution can influence model fit.     

Host-parasitoid dynamics of a generalized Thompson model

A discrete-time host-parasitoid model including host-density dependence and a generalized Thompson escape function is analyzed. This model assumes that parasitoids are egg-limited but not search-limited, and is proven to exhibit five types of dynamics: host failure in which the host goes extinct in the parasitoid's presence or absence, unconditional parasitoid failure in which the parasitoid always goes extinct while the host persists, conditional parasitoid failure in the host and the parasitoid go extinct or coexist depending on the initial host-parasitoid ratio, parasitoid driven extinction in which the parasitoid invariably drives the host to extinction, and coexistence in which the host and parasitoid coexist about a global attractor. The latter two dynamics only occur when the parasitoid's maximal rate of growth exceeds the host's maximal rate of growth. Moreover, coexistence requires parasitism events to be sufficiently aggregated. Small additive noise is proven to alter the dynamical outcomes in two ways. The addition of noise to parasitoid driven extinction results in random outbreaks of the host and parasitoid with varying intensity. Additive noise converts conditional parasitoid failure to unconditional parasitoid failure. Implications for classical biological control are discussed.     

Population outbreaks in a discrete world

We present and analyze a simple three-patch host-parasitoid model where population growth is discrete. The model gives solutions that are qualitatively similar to the stable large-amplitude patterns in space found in reaction-diffusion theory. In the context of host-parasitoid interactions, the large-amplitude portions of the solution can be thought of as spatially localized host population outbreaks. Here, we show that the biological requirements for localized population outbreaks in a discrete world are identical to those found in reaction- diffusion theory. Furthermore, the model conveniently allows investigation into the robustness of these population outbreaks under the influence of density-dependent dispersal behavior. We find that localized population outbreaks in space can still occur with modest amounts of pursuit and aggregative behavior by parasitoids. We end by showing that evidence from a real host-parasitoid system is consistent with the predictions of the model.     

Dynamical consequences of optimal host feeding on host-parasitoid population dynamics

This study examines the influence of various host-feeding patterns on host-parasitoid population dynamics. The following types of host-feeding patterns are considered: concurrent and non-destructive, non-concurrent and non-destructive, and non-concurrent and destructive. The host-parasitoid population dynamics is described by the Lotka-Volterra continuous-time model. This study shows that when parasitoids behave optimally, i.e. they maximize their fitness measured by the instantaneous per capita growth rate, the non-destructive type of host feeding stabilizes host-parasitoid dynamics. Other types of host feeding, i.e. destructive, concurrent, or non-concurrent, do not qualitatively change the neutral stability of the Lotka-Volterra model. Moreover, it is shown that the pattern of host feeding which maximizes parasitoid fitness is either non-concurrent and destructive, or concurrent and non-destructive host feeding, depending on the host abundance and parameters of the model. The effects of the adaptive choice of host-feeding patterns on host-parasitoid population dynamics are discussed.     

Variables Aggregation in Time Varying Discrete Systems

In this work we extend approximate aggregation methods in time discrete linear models to the case of time varying environments. Approximate aggregation consists in describing some features of the dynamics of a general system involving many coupled variables in terms of the dynamics of a reduced system with a few number of variables. We present a time discrete time varying model in which we distinguish two time scales. By using perturbation methods we transform the system to make the global variables appear and build up the aggregated system. The asymptotic relationships between the general and aggregated systems are explored in the cases of a cyclically varying environment and a changing environment in process of stabilization. We show that under quite general conditions the knowledge of the behavior of the aggregated system characterizes that of the general system. The general method is also applied to aggregate a multiregional time dependent Leslie model showing that the aggregated model has demographic rates depending on the equilibrium proportions of individuals in the different patches.     

Differential cannibalism and population dynamics in a host-parasitoid system

The effects of host cannibalism on a host-parasitoid system were explored through experiment and modelling. In individual encounters between parasitized and unparasitized Plodia interpunctella larvae, parasitized larvae were more likely to be cannibalized. Inclusion of this differential cannibalism into a simple Lotka-Volterra-type model of host-parasitoid population dynamics generates alternative stable states-including stable coexistence and extinction of the parasitoid — which depend on starting conditions. Possible mechanisms for differential cannibalism, and its implications for studies of host-parasitoid populations and biological control programmes are discussed.     

Strange limits of stability in host-parasitoid systems

The classical Nicholson-Bailey model for a two species host-parasitoid system with discrete generations assumes random distributions of both hosts and parasitoids, randomly searching parasitoids, and random encounters between the individuals of the two species. Although unstable, this model induced many investigations into more complex host-parasitoid systems. Local linearized stability analysis shows that equilibria of host parasitoid systems within the framework of a generalized Nicholson-Bailey model are generally unstable. Stability is only possible if host fertility does not exceede ⁴=54.5982 and if superparasitism is unsuccessful. This special situation has already been discovered by Hassell et al. (1983) in their study of the effects of variable sex ratios on host parasitoid dynamics. We discuss global behaviour of the Hassell-Waage-May model using KAM-theory and illustrate its sensitivity to small perturbations, which can give rise to radically different patterns of the population dynamics of interacting hosts and parasitoids.     

The effect of evolution on host-parasitoid systems

It is well known that a simple first-order difference equation can exhibit complex population dynamics, such as sustained oscillations and chaos. An interesting problem is whether such oscillatory dynamics are expected to occur in real populations. This paper assumes that the resident system is composed of 1-host and 1-parasitoid and that only the host is allowed to evolve, but not the parasitoid. Based on the invasibility of a host to host-parasitoid systems, we investigate the dynamics of the host-parasitoid system favored by natural selection. We consider two cases. In the first case, the host's evolution involving both the intrinsic growth rate and the sensitivity to density is considered. In the second case, the host's evolution involving both the intrinsic growth rate and the vulnerability to the parasitoid is considered. In both cases, we see that the dynamics with a stable equilibrium will not be favored by natural selection without the trade-off between the host's traits which are allowed to evolve. The host-parasitoid system with a stable equilibrium will be eventually invaded by a host type that develops an unstable equilibrium with the parasitoid. If there is a trade-off between the host's traits which are allowed to evolve, a host-parasitoid system with a stable equilibrium can be favored by natural selection.