Chemotaxis-induced spatio-temporal heterogeneity in multi-species host-parasitoid systems

When searching for hosts, parasitoids are observed to aggregate in response to chemical signalling cues emitted by plants during host feeding. In this paper we model aggregative parasitoid behaviour in a multi-species host-parasitoid community using a system of reaction-diffusion-chemotaxis equations. The stability properties of the steady-states of the model system are studied using linear stability analysis which highlights the possibility of interesting dynamical behaviour when the chemotactic response is above a certain threshold. We observe quasi-chaotic dynamic heterogeneous spatio-temporal patterns, quasi-stationary heterogeneous patterns and a destabilisation of the steady-states of the system. The generation of heterogeneous spatio-temporal patterns and destabilisation of the steady state are due to parasitoid chemotactic response to hosts. The dynamical behaviour of our system has both mathematical and ecological implications and the concepts of chemotaxis-driven instability and coexistence and ecological change are discussed. I. G. Pearce gratefully acknowledges the financial support of the NERC.     

Spatio-temporal patterns of gene flow and dispersal under temperature increase

Recent data show that the Earth climate is undergoing a change at a rate which is outstanding in geologic history. Temperature is one of the major driving forces of gene flow and dispersal. In this paper the spatial dynamics of genetic dispersal is studied under the auspices of temperature increase by means of a mathematical model. The main elements genetics, competition and dispersal are combined in a coherent approach by a system of coupled partial differential equations with non-linear reaction terms describing population dynamics, genetic exchange and competition. Temperature reaction norms are conferred by a two allele system. The non-linearities of the interaction terms give rise to a richness of spatio-temporal dynamic patterns. Here we show how invasion processes in form of travelling waves are initiated by a temperature rise.     

Modelling the effect of temperature on the range expansion of species by reaction-diffusion equations

The spatial dynamics of range expansion is studied in dependence of temperature. The main elements population dynamics, competition and dispersal are combined in a coherent approach based on a system of coupled partial differential equations of the reaction-diffusion type. The nonlinear reaction terms comprise population dynamic models with temperature dependent reproduction rates subject to an Allee effect and mutual competition. The effect of temperature on travelling wave solutions is investigated for a one dimensional model version. One main result is the importance of the Allee effect for the crossing of regions with unsuitable habitats. The nonlinearities of the interaction terms give rise to a richness of spatio-temporal dynamic patterns. In two dimensions, the resulting non-linear initial boundary value problems are solved over geometries of heterogeneous landscapes. Geo referenced model parameters such as mean temperature and elevation are imported into the finite element tool COMSOL Multiphysics from a geographical information system. The model is applied to the range expansion of species at the scale of middle Europe.     

On the determinants of population structure in antigenically diverse pathogens

Many pathogens exhibit antigenic diversity and elicit strain-specific immune responses. This potential for cross-immunity structure in the host resource motivates the development of mathematical models, stressing competition for susceptible hosts in driving pathogen population dynamics and genetics. Here we establish that certain model formulations exhibit characteristics of prototype pattern-forming systems, with pathogen population structure emerging as three possible patterns: (i) incidence is steady and homogeneous; (ii) incidence is steady but heterogeneous; and (iii) incidence shows oscillatory dynamics, with travelling waves in strain-space. Results are robust to strain number, but sensitive to the mechanism of cumulative immunity.     

Travelling waves of a delayed SIR epidemic model with nonlinear incidence rate and spatial diffusion

This paper is concerned with the existence of travelling waves to a SIR epidemic model with nonlinear incidence rate, spatial diffusion and time delay. By analyzing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state to this system under homogeneous Neumann boundary conditions is discussed. By using the cross iteration method and the Schauder's fixed point theorem, we reduce the existence of travelling waves to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derive the existence of a travelling wave connecting the disease-free steady state and the endemic steady state. Numerical simulations are carried out to illustrate the main results.     

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.     

Dynamics of Notch Activity in a Model of Interacting Signaling Pathways

Networks of interacting signaling pathways are formulated with systems of reaction-diffusion (RD) equations. We show that weak interactions between signaling pathways have negligible effects on formation of spatial patterns of signaling molecules. In particular, a weak interaction between Retinoic Acid (RA) and Notch signaling pathways does not change dynamics of Notch activity in the spatial domain. Conversely, large interactions of signaling pathways can influence effects of each signaling pathway. When the RD system is largely perturbed by RA-Notch interactions, new spatial patterns of Notch activity are obtained. Moreover, analysis of the perturbed Homogeneous System (HS) indicates that the system admits bifurcating periodic orbits near a Hopf bifurcation point. Starting from a neighborhood of the Hopf bifurcation, oscillatory standing waves of Notch activity are numerically observed. This is of particular interest since recent laboratory experiments confirm oscillatory dynamics of Notch activity.     

Coexistence of multiple parasitoids on a single host due to differences in parasitoid phenology

There are many well-documented cases in which multiple parasitoids can coexist on a single host species. We examine a theoretical framework to assess whether parasitoid coexistence can be explained through differences in timing of parasitoid oviposition and parasitoid emergence. This study explicitly includes the phenology of host and parasitoid development and explores how this mechanism affects the population dynamics. Coexistence of the host with two parasitoids requires a balance between parasitoid fecundity and survival and occurs most readily if one parasitoid attacks earlier but emerges later than the other parasitoid. The host density can either be decreased or increased when a second coexisting parasitoid is introduced into the system. However, there always exists a single parasitoid type that is most effective at depressing the host density, although this type may not be successful due to parasitoid competition. The coexistence of multiple parasitoids also affects the population dynamics. For instance, population oscillations can be removed by the introduction of a second parasitoid. In general, subtle differences in parasitoid phenology can give rise to different outcomes in a host–multi-parasitoid system, and this may offer some insight into why establishing criteria for the ‘ideal’ biological control agent has been so challenging.     

Spatially mismatched trophic dynamics: cyclically outbreaking geometrids and their larval parasitoids

For trophic interactions to generate population cycles and complex spatio-temporal patterns, like travelling waves, the spatial dynamics must be matched across trophic levels. Here, we propose a spatial methodological approach for detecting such spatial match–mismatch and apply it to geometrid moths and their larval parasitoids in northern Norway, where outbreak cycles and travelling waves occur. We found clear evidence of spatial mismatch, suggesting that the spatially patterned moth cycles in this system are probably ruled by trophic interactions involving other agents than larval parasitoids.     

Generation of periodic waves by landscape features in cyclic predator-prey systems

The vast majority of models for spatial dynamics of natural populations assume a homogeneous physical environment. However, in practice, dispersing organisms may encounter landscape features that significantly inhibit their movement. We use mathematical modelling to investigate the effect of such landscape features on cyclic predator-prey populations. We show that when appropriate boundary conditions are applied at the edge of the obstacle, a pattern of periodic travelling waves develops, moving out and away from the obstacle. Depending on the assumptions of the model, these waves can take the form of roughly circular 'target patterns' or spirals. This is, to our knowledge, a new mechanism for periodic-wave generation in ecological systems and our results suggest that it may apply quite generally not only to cyclic predator-prey interactions, but also to populations that oscillate for other reasons. In particular, we suggest that it may provide an explanation for the observed pattern of travelling waves in the densities of field voles (Microtus agrestis) in Kielder Forest (Scotland-England border) and of red grouse (Lagopus lagopus scoticus) on Kerloch Moor (northeast Scotland), which in both cases move orthogonally to any large-scale obstacles to movement. Moreover, given that such obstacles to movement are the rule rather than the exception in real-world environments, our results suggest that complex spatio-temporal patterns such as periodic travelling waves are likely to be much more common in the natural world than has previously been assumed.     

Stability switches,oscillatory multistability,and spatio-temporal patterns of nonlinear oscillations in recurrently delay coupled neural networks

A model of time-delay recurrently coupled spatially segregated neural assemblies is here proposed. We show that it operates like some of the hierarchical architectures of the brain. Each assembly is a neural network with no delay in the local couplings between the units. The delay appears in the long range feedforward and feedback inter-assemblies communications. Bifurcation analysis of a simple four-units system in the autonomous case shows the richness of the dynamical behaviors in a biophysically plausible parameter region. We find oscillatory multistability, hysteresis, and stability switches of the rest state provoked by the time delay. Then we investigate the spatio-temporal patterns of bifurcating periodic solutions by using the symmetric local Hopf bifurcation theory of delay differential equations and derive the equation describing the flow on the center manifold that enables us determining the direction of Hopf bifurcations and stability of the bifurcating periodic orbits. We also discuss computational properties of the system due to the delay when an external drive of the network mimicks external sensory input.     

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.     

Spatial and spatio-temporal patterns in a cell-haptotaxis model

We investigate a cell-haptotaxis model for the generation of spatial and spatio-temporal patterns in one dimension. We analyse the steady state problem for specific boundary conditions and show the existence of spatially hetero-geneous steady states. A linear analysis shows that stability is lost through a Hopf bifurcation. We carry out a nonlinear multi-time scale perturbation procedure to study the evolution of the resulting spatio-temporal patterns. We also analyse the model in a parameter domain wherein it exhibits a singular dispersion relation.     

Travelling wave dynamics and self-organization in a spatio-temporally structured population

We analyse spatial population dynamics showing that periodic or period-like chaotic dynamics produce self-organization structures, such as travelling waves. We suggest that self-organized patterns are associated with spatial synchrony patterns that often depend on geographical distance between subpopulations. The population dynamics also show statistical spatial autocorrelation patterns. We contrast our theoretical simulations with empirical data on annual damages in young sapling stands caused by voles. We conclude, on the basis of the periodicity, synchrony, and spatial autocorrelation patterns, and our simulation results, that vole dynamics represent travelling waves in population dynamics. We suggest that because such synchrony patterns are frequently observed in natural populations, spatial self-organization may be more common in population dynamics than reported in the literature.     

Discrete-time travelling waves: Ecological examples

Integrodifference equations are discrete-time models that possess many of the attributes of continuous-time reaction-diffusion equations. They arise naturally in population biology as models for organisms with discrete nonoverlapping generations and well-defined growth and dispersal stages. I examined the varied travelling waves that arise in some simple ecologically-interesting integrodifference equations. For a scalar equation with compensatory growth, I observed only simple travelling waves. For carefully chosen redistribution kernels, one may derive the speed and approximate the shape of the observed waveforms. A model with overcompensation exhibited flip bifurcations and travelling cycles in addition to simple travelling waves. Finally, a simple predator-prey system possessed periodic wave trains and a variety of travelling waves.     

Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations

We investigate the emergence of spatio-temporal patterns in ecological systems. In particular, we study a generalized predator-prey system on a spatial domain. On this domain diffusion is considered as the principal process of motion. We derive the conditions for Hopf and Turing instabilities without specifying the predator-prey functional responses and discuss their biological implications. Furthermore, we identify the codimension-2 Turing-Hopf bifurcation and the codimension-3 Turing-Takens-Bogdanov bifurcation. These bifurcations give rise to complex pattern formation processes in their neighborhood. Our theoretical findings are illustrated with a specific model. In simulations a large variety of different types of long-term behavior, including homogenous distributions, stationary spatial patterns and complex spatio-temporal patterns, are observed.     

Phase resetting and bifurcation in the ventricular myocardium.

With the dynamic differential equations of Beeler, G. W., and H. Reuter (1977, J. Physiol. [Lond.]. 268:177-210), we have studied the oscillatory behavior of the ventricular muscle fiber stimulated by a depolarizing applied current I app. The dynamic solutions of BR equations revealed that as I app increases, a periodic repetitive spiking mode appears above the subthreshold I app, which transforms to a periodic spiking-bursting mode of oscillations, and finally to chaos near the suprathreshold I app (i.e., near the termination of the periodic state). Phase resetting and annihilation of repetitive firing in the ventricular myocardium were demonstrated by a brief current pulse of the proper magnitude applied at the proper phase. These phenomena were further examined by a bifurcation analysis. A bifurcation diagram constructed as a function of I app revealed the existence of a stable periodic solution for a certain range of current values. Two Hopf bifurcation points exist in the solution, one just above the lower periodic limit point and the other substantially below the upper periodic limit point. Between each periodic limit point and the Hopf bifurcation, the cell exhibited the coexistence of two different stable modes of operation; the oscillatory repetitive firing state and the time-independent steady state. As in the Hodgkin-Huxley case, there was a low amplitude unstable periodic state, which separates the domain of the stable periodic state from the stable steady state. Thus, in support of the dynamic perturbation methods, the bifurcation diagram of the BR equation predicts the region where instantaneous perturbations, such as brief current pulses, can send the stable repetitive rhythmic state into the stable steady state.     

The logistic equation with a diffusionally coupled delay

The asymptotic behaviour of a logistic equation with diffusion on a bounded region and a diffusionally coupled delay is investigated. An equivelent parabolic system is derived for certain types of delays. Using a Layapunov functional, sufficient conditions for the global asymptotic stability of the constant steady state are obtained. When the global stability is lost, using Hopf's bifurcation theory, existence of travelling waves is shown for ring-like and periodic one dimensional habitats.