A Mathematical Biologist’s Guide to Absolute and Convective Instability

Mathematical models have been highly successful at reproducing the complex spatiotemporal phenomena seen in many biological systems. However, the ability to numerically simulate such phenomena currently far outstrips detailed mathematical understanding. This paper reviews the theory of absolute and convective instability, which has the potential to redress this inbalance in some cases. In spatiotemporal systems, unstable steady states subdivide into two categories. Those that are absolutely unstable are not relevant in applications except as generators of spatial or spatiotemporal patterns, but convectively unstable steady states can occur as persistent features of solutions. The authors explain the concepts of absolute and convective instability, and also the related concepts of remnant and transient instability. They give examples of their use in explaining qualitative transitions in solution behaviour. They then describe how to distinguish different types of instability, focussing on the relatively new approach of the absolute spectrum. They also discuss the use of the theory for making quantitative predictions on how spatiotemporal solutions change with model parameters. The discussion is illustrated throughout by numerical simulations of a model for river-based predator–prey systems.     

Instabilities and spatiotemporal patterns behind predator invasions with nonlocal prey competition

We study the influence of nonlocal intraspecies prey competition on the spatiotemporal patterns arising behind predator invasions in two oscillatory reaction–diffusion integro-differential models. We use three common types of integral kernels as well as develop a caricature system, to describe the influence of the standard deviation and kurtosis of the kernel function on the patterns observed. We find that nonlocal competition can destabilize the spatially homogeneous state behind the invasion and lead to the formation of complex spatiotemporal patterns, including stationary spatially periodic patterns, wave trains and irregular spatiotemporal oscillations. In addition, the caricature system illustrates how large standard deviation and low kurtosis facilitate the formation of these spatiotemporal patterns. This suggests that nonlocal competition may be an important mechanism underlying spatial pattern formation, particularly in systems where the competition between individuals varies over space in a platykurtic manner.     

Instabilities and spatiotemporal patterns behind predator invasions with nonlocal prey competition

We study the influence of nonlocal intraspecies prey competition on the spatiotemporal patterns arising behind predator invasions in two oscillatory reaction–diffusion integro-differential models. We use three common types of integral kernels as well as develop a caricature system, to describe the influence of the standard deviation and kurtosis of the kernel function on the patterns observed. We find that nonlocal competition can destabilize the spatially homogeneous state behind the invasion and lead to the formation of complex spatiotemporal patterns, including stationary spatially periodic patterns, wave trains and irregular spatiotemporal oscillations. In addition, the caricature system illustrates how large standard deviation and low kurtosis facilitate the formation of these spatiotemporal patterns. This suggests that nonlocal competition may be an important mechanism underlying spatial pattern formation, particularly in systems where the competition between individuals varies over space in a platykurtic manner.     

Spatiotemporal complexity of patchy invasion in a predator-prey system with the Allee effect

Invasion of an exotic species initiated by its local introduction is considered subject to predator-prey interactions and the Allee effect when the prey growth becomes negative for small values of the prey density. Mathematically, the system dynamics is described by two nonlinear diffusion-reaction equations in two spatial dimensions. Regimes of invasion are studied by means of extensive numerical simulations. We show that, in this system, along with well-known scenarios of species spread via propagation of continuous population fronts, there exists an essentially different invasion regime which we call a patchy invasion. In this regime, the species spreads over space via irregular motion and interaction of separate population patches without formation of any continuous front, the population density between the patches being nearly zero. We show that this type of the system dynamics corresponds to spatiotemporal chaos and calculate the dominant Lyapunov exponent. We then show that, surprisingly, in the regime of patchy invasion the spatially average prey density appears to be below the survival threshold. We also show that a variation of parameters can destroy this regime and either restore the usual invasion scenario via propagation of continuous fronts or brings the species to extinction; thus, the patchy spread can be qualified as the invasion at the edge of extinction. Finally, we discuss the implications of this phenomenon for invasive species management and control.     

Patterns of Patchy Spread in Deterministic and Stochastic Models of Biological Invasion and Biological Control

A few spatiotemporal models of population dynamics are considered in relation to biological invasion and biological control. The patterns of spread in one and two spatial dimensions are studied by means of extensive numerical simulations. We show that, in the case that population multiplication is damped by the strong Allee effect (when the population growth rate becomes negative for small population density), in a certain parameter range the spread can take place not via the intuitively expected circular expanding population front but via motion and interaction of separate patches. Alternatively, the patchy spread can take place in a system without Allee effect as a result of strong environmental noise. We then show that the phenomenon of deterministic patchy invasion takes place ‘at the edge of extinction’ so that a small change of controlling parameters either brings the species to extinction or restores the travelling population fronts. Moreover, we show that the regime of patchy invasion in two spatial dimensions actually takes place when the species go extinct in the corresponding 1-D system.     

Cell proliferation drives neural crest cell invasion of the intestine

A general mathematical model of cell invasion is developed and validated with an experimental system. The model incorporates two basic cell functions: non-directed (diffusive) motility and proliferation to a carrying capacity limit. The model is used here to investigate cell proliferation and motility differences along the axis of an invasion wave. Mathematical simulations yield surprising and counterintuitive predictions. In this general scenario, cells at the invasive front are proliferative and migrate into previously unoccupied tissues while those behind the front are essentially nonproliferative and do not directly migrate into unoccupied tissues. These differences are not innate to the cells, but are a function of proximity to uninvaded tissue. Therefore, proliferation at the invading front is the critical mechanism driving apparently directed invasion. An appropriate system to experimentally validate these predictions is the directional invasion and colonization of the gut by vagal neural crest cells that establish the enteric nervous system. An assay using gut organ culture with chick-quail grafting is used for this purpose. The experimental results are entirely concordant with the mathematical predictions. We conclude that proliferation at the wavefront is a key mechanism driving the invasive process. This has important implications not just for the neural crest, but for other invasion systems such as epidermal wound healing, carcinoma invasion and other developmental cell migrations.     

Periodic travelling waves in cyclic predator–prey systems

Predation is an established cause of cycling in prey species. Here, the ability of predation to explain periodic travelling waves in prey populations, which have recently been found in a number of spatiotemporal field studies, is examined. The nature of periodic waves in these systems, and the way in which they can be generated by the invasion of predators into a prey population is discussed. A theoretical calculation that predicts, as a function of two parameter ratios, whether such an invasion will lead to a stable periodic travelling wave that would be observed in practice is presented ‐ the alternative outcome is spatiotemporal chaos. The calculation also predicts quantitative details of the periodic waves, such as speed and amplitude. The results give new insights into the types of predator‐prey systems in which one would expect to see periodic travelling waves following an invasion by predators.     

Oscillations and chaos behind predator-prey invasion: mathematical artifact or ecological reality?

A constant dilemma in theoretical ecology is knowing whether model predictions corrspond to real phenomena or whether they are artifacts of the modelling framework. The frequent absence of detailed ecological data against which models can be tested gives this issue particular importance. We address this question in the specific case of invasion in a predator-prey system with oscillatory population kinetics, in which both species exhibit local random movement. Given only these two basic qualitative features, we consider whether we can deduce any properties of the behaviour following invasion. To do this we study four different types of mathematical model, which have no formal relationship, but which all reflect our two qualitative ingredients. The models are: reaction-diffusion equations, coupled map lattices, deterministic cellular automata, and integrodifference equations. We present results of numerical simulations of the invasion of prey by predators for each model, and show that although there are certain differences, the main qualitative features of the behaviour behind invasion are the same for all the models. Specifically, there are either irregular spatiotemporal oscillations behind the invasion, or regular spatiotemporal oscillations with the form of a periodic travelling ''wake'', depending on parameter values. The observation of this behaviour in all types of model strongly suggests that it is a direct consequence of our basic qualitative assumptions, and as such is an ecological reality which will always occur behind invasion in actual oscillatory predator-prey systems.     

Collective behavior in cancer cell populations

In recent years the argument has been made that malignant tumors represent complex dynamic and self‐organizing biosystems. Furthermore, there is increasing evidence that collective cell migration is common during invasion and metastasis of malignant tumors. Here, we argue that cancer systems may be capable of developing multicellular collective patterns that resemble evolved adaptive behavior known from other biological systems including collective sensing of environmental conditions and collective decision‐making. We present a concept as to how these properties could arise in tumors and why the emergence of such swarm‐like patterns would confer advantageous properties to the spatiotemporal expansion of tumors, and consequently, why understanding and ultimately targeting such collectivity should be of interest for basic and clinical cancer research alike.     

Consistent life history shifts along invasion routes? An examination of round goby populations invading on two continents

Aim

Many invasive populations exhibit dynamic life history shifts along their invasion route. We investigated whether these shifts represent consistent biological responses of a given species to range expansion, even in systems located in different geographic regions.

Location

North‐eastern France, Central Ontario (Canada).

Method

We investigated population density, life history traits and age‐specific reproductive investment in expanding populations of round goby at three invasion stages (expansion front, area colonized one year earlier and area colonized for ~five years) along the invasive routes in two river systems differing in climate and system productivity. Interindividual variability, shown to affect range expansion rates, was also investigated along the invasion routes. The study was based on female round gobies collected in three locations within each invasion stage twice monthly throughout the reproductive season (March/May to July).

Results

In both systems, reproductive investment was highest in the newly colonized area and decreased with time since colonization. A faster decrease in reproductive investment was found in the warmer, more productive system behind the invasion front, potentially associated with faster population growth and increased intraspecific competition. In both systems, individual variability in growth and reproductive traits increased from the newly colonized area to the areas of earlier colonization.

Main conclusions

The patterns observed in the two systems suggest a common invasion strategy independent of environmental conditions and highlight the dynamic nature of invasive populations’ life history behind the invasion front. Common energetic allocation strategies can be expected at the invasion front. Range expansion may be associated with population growth induced by rapid acclimation to biotic conditions associated with range shift.

     

Home-loving boreal hare mitochondria survived several invasions in Iberia: the relative roles of recurrent hybridisation and allele surfing

Genetic introgression from a resident species into an invading close relative can result from repeated hybridisation along the invasion front and/or allele surfing on the expansion wave. Cases where the phenomenon is massive and systematic, such as for hares (genus Lepus) in Iberia, would be best explained by recurrent hybridisation but this is difficult to prove because the donor populations are generally extinct. In the Pyrenean foothills, Lepus europaeus presumably replaced Lepus granatensis recently and the present species border is parallel to the direction of invasion, so that populations of L. granatensis in the contact zone represent proxies of existing variation before the invasion. Among three pairs of populations sampled across this border, we find less differentiation of mitochondrial DNA (mtDNA) across than along it, as predicted under recurrent hybridisation at the invasion front. Using autosomal microsatellite loci and X- and Y-linked diagnostic loci, we show that admixture across the border is quasi-absent, making it unlikely that lack of interspecific mtDNA differentiation results from ongoing gene flow. Furthermore, we find that the local species ranges are climatically contrasted, making it also unlikely that ongoing ecology-driven movement of the contact account for mtDNA introgression. The lack of mtDNA differentiation across the boundary is mostly due to sharing of mtDNA from a boreal species currently extinct in Iberia (Lepus timidus) whose mitochondria have thus remained in place since the last deglaciation despite successive invasions by two other species. Home-loving mitochondria thus witness past species distribution rather than ongoing exchanges across stabilised contact zones.     

Capturing the Dynamics of a Hybrid Multiscale Cancer Model with a Continuum Model

Cancer is a complex disease involving processes at spatial scales from subcellular, like cell signalling, to tissue scale, such as vascular network formation. A number of multiscale models have been developed to study the dynamics that emerge from the coupling between the intracellular, cellular and tissue scales. Here, we develop a continuum partial differential equation model to capture the dynamics of a particular multiscale model (a hybrid cellular automaton with discrete cells, diffusible factors and an explicit vascular network). The purpose is to test under which circumstances such a continuum model gives equivalent predictions to the original multiscale model, in the knowledge that the system details are known, and differences in model results can be explained in terms of model features (rather than unknown experimental confounding factors). The continuum model qualitatively replicates the dynamics from the multiscale model, with certain discrepancies observed owing to the differences in the modelling of certain processes. The continuum model admits travelling wave solutions for normal tissue growth and tumour invasion, with similar behaviour observed in the multiscale model. However, the continuum model enables us to analyse the spatially homogeneous steady states of the system, and hence to analyse these waves in more detail. We show that the tumour microenvironmental effects from the multiscale model mean that tumour invasion exhibits a so-called pushed wave when the carrying capacity for tumour cell proliferation is less than the total cell density at the tumour wave front. These pushed waves of tumour invasion propagate by triggering apoptosis of normal cells at the wave front. Otherwise, numerical evidence suggests that the wave speed can be predicted from linear analysis about the normal tissue steady state.     

Metapopulation dynamics and biological invasions: a spatially explicit model applied to a freshwater snail

The spatial spread of invading organisms is a major contemporary concern. We focus here on invasions in inherently fragmented habitats, such as freshwater systems, and explore the usefulness of metapopulation models in this context. Maximum-likelihood methods allow the estimation of colonization and extinction rates, as functions of habitat patch sizes and positions, from time series of presence/absence data. This framework also provides confidence intervals of these estimates and hypotheses tests. We analyze a previously unpublished 12-year survey of the spread of the introduced snail Tarebia granifera in 47 Martinican rivers. Simple metapopulation models reproduce with reasonable accuracy several quantitative aspects of the invasion, including regional abundance, spatiotemporal structure, and site-by-site colonization dates. Sensitivity analysis reveals that the invasion sequence depended strongly on metapopulation size (number of sites) and spatial structure (distances among sites). The invasion history has also been accelerated by stochastic events, as illustrated by a large, central river that happened to be colonized very early and served as an invasion pool. Finally, we discuss the benefits of this approach for the understanding of invasions in fragmented landscapes.     

Spatial interplay of plant competition and consumer foraging mediate plant coexistence and drive the invasion ratchet

Indirect effects may play an important role in structuring plant communities. Using a spatially explicit model of consumer foraging and plant competition, we demonstrate how the relationship between the spatial area over which plants compete and the spatial scale of consumer behaviour can determine the outcome of competition when one plant species provides a refuge for mobile consumers (i.e. refuge-mediated apparent competition). Once an initial population of the invader is established, complete invasion may be inevitable because of an ever-advancing invasion front ratchets forward driven by a feeding front of mobile consumers. Because the spatial extent of apparent competition determines the area available for colonization, consumers may also dictate the rate at which an invasion occurs. We find that, as long as refuge-mediated apparent competition is sufficiently localized, invasion is possible even in systems characterized by low overall levels of consumer pressure. Moreover, we show that a stable equilibrium can result in which both resident and invading plants coexist, suggesting that spatial heterogeneity created by refuge-mediated apparent competition may be important in mediating coexistence in plant communities. The spatial interplay of consumer behaviour and plant competition may be an underappreciated mechanism affecting the composition, diversity and spatial pattern of plant communities.     

Stress and immunity at the invasion front: a comparison across cane toad (Rhinella marina) populations

At an invasion front, energetic and physiological trade‐offs may differ from those at the range‐core as a result of selection for enhanced dispersal, combined with a low density of conspecifics (which reduces pathogen transmission and competition for food). We measured traits related to energy stores and immunity in wild cane toads (Rhinella marina) across a 750‐km transect from their invasion front in tropical Australia, back into sites colonized 21 years earlier. Several traits were found to vary with population age; some linearly and others in a curvilinear manner. The relative size of spleens and fat bodies was highest in the oldest and newest populations, where rates of lungworm infection were lowest. Toads from older populations produced more corticosterone in response to a standardized stressor, and had higher lymphocyte counts (but lower basophil counts). The amount of skin swelling elicited by phytohaemagglutinin injection did not vary geographically, although recruitment of leukocytes to the injected tissue was higher in toads from long‐colonized areas. Because this was a field‐based study, we cannot differentiate the effects of population age, toad density or pathogen pressure on our measures of stress and immune responses, nor can we distinguish whether the causation involves hard‐wired adaptive processes or phenotypically plastic responses. Nonetheless, our data demonstrate substantial variation in immune systems among toads at varying distances from an invasion front, showing that a biological invasion imposes strong pressures on physiological systems of the invader.     

Reduced investment in immune function in invasion-front populations of the cane toad (<Emphasis Type="Italic">Rhinella marina</Emphasis>) in Australia

In an invasive species, selection for increased rates of dispersal at the expanding range front may favor the evolution of reduced investment into any trait that does not contribute to more rapid dispersal. Thus, populations at the invasion front may exhibit reduced investment into the immune system. To test this prediction, cane toads (Rhinella marina) from parents collected from populations across the toads’ invasion history in tropical Australia were raised in a standard environment. When their immune systems were challenged by injection of bacterial lipopolysaccharide, the toads’ metabolic rates rose by up to 40%. The magnitude of elevation in metabolic rate was lower in toads derived from the invasion front than in those from long-established populations. Our results support the hypothesis that an animal’s investment in immune defenses can be modified by selective forces that arise in the course of a biological invasion.     

Self-consistent model of a pulsed air discharge excited by surface waves

A complete self-consistent electrodynamic model of a pulsed gas discharge excited by surface waves is developed. The model allows one to calculate both the initial phase of the discharge front propagation and the parameters of the produced plasma. The spatiotemporal evolution of the electromagnetic field and plasma parameters at the discharge front is investigated for the first time. It is shown that discharge propagation is mainly governed by a breakdown wave in an inhomogeneous electric field at the leading edge of the ionization front. It is found that the effect of the electric field enhancement in the plasma resonance region significantly affects the velocity of the breakdown wave. The results of calculations agree well with experimental data.     

Chiral symmetry-breaking transition in growth front of crystal phase of 1,1'-binaphthyl in its supercooled melt

Although the theory of spontaneous chiral symmetry-breaking in open systems was proposed some time ago, experimental realization of this phenomenon has not been achieved. In this article, we note that the crystal growth front of 1,1'-binaphthyl shows many of the characteristics of an open system in which chiral symmetry-breaking has occurred. We studied the temperature profiles of the crystallizing surface and obtained X-ray diffraction data of the crystals grown from the melt under controlled conditions. The data show that, under appropriate conditions, the observed bimodal probability distribution of enantiomeric excess (ee) with maxima approximately 60% is due exclusively to chiral crystals and not due to racemic crystals of 1,1'-binaphthyl that can also form at large supercooling. The mass independence of the ee shows that the growing front maintains a constant ee, which is a clear signature of open systems in steady state. Chirality 16:131-136, 2004.     

Speed of invasion in lattice population models: pair-edge approximation

 We propose a simple approach to approximating the speed of invasion in lattice population models. Approximate critical parameter values for successful invasion are then found by solving for zero wave speed. The approximation is based on describing the occupied region by the ordinary pair approximation, and using quasi-steady-state pair approximations to describe the leading edge of the wave front. We illustrate this idea using the basic contact process on the 1 and 2 dimensional lattice (with and without nearest-neighbor migration), finding very good agreement between the approximation and simulation results. The approximate critical values obtained by our approximation are significantly more accurate than those obtained by the ordinary pair approximation. Received 4 September 1996     

Secretion of type III effectors into host cells in real time

Type III secretion (T3S) systems are key features of many gram-negative bacteria that translocate T3S effector proteins directly into eukaryotic cells. There, T3S effectors exert many effects, such as cellular invasion or modulation of host immune responses. Studying spatiotemporal orchestrated secretion of various effectors has been difficult without disrupting their functions. Here we developed a new approach using Shigella flexneri T3S as a model to investigate bacterial translocation of individual effectors via multidimensional time-lapse microscopy. We demonstrate that direct fluorescent labeling of tetracysteine motif-tagged effectors IpaB and IpaC is possible in situ without loss of function. Studying the T3S kinetics of IpaB and IpaC ejection from individual bacteria, we found that the entire pools of IpaB and IpaC were released concurrently upon host cell contact, and that 50% of each effector was secreted in 240 s. This method allows an unprecedented analysis of the spatiotemporal events during T3S.