Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations

 In this paper we study the existence of one-dimensional travelling wave solutions u(x, t)=φ(x−ct) for the non-linear degenerate (at u=0) reaction-diffusion equation u _t =[D(u)u _x ] _x +g(u) where g is a generalisation of the Nagumo equation arising in nerve conduction theory, as well as describing the Allee effect. We use a dynamical systems approach to prove: 1. the global bifurcation of a heteroclinic cycle (two monotone stationary front solutions), for c=0, 2. The existence of a unique value c ^*>0 of c for which φ(x−c ^* t) is a travelling wave solution of sharp type and 3. A continuum of monotone and oscillatory fronts for c≠c ^*. We present some numerical simulations of the phase portrait in travelling wave coordinates and on the full partial differential equation. Received 15 December 1995; received in revised form 14 May 1996     

Overcompensatory recruitment and generation delay in discrete age-structured population models

 The effect of overcompensatory recruitment and the combined effect of overcompensatory recruitment and generation delay in discrete nonlinear age-structured population models is studied. Considering overcompensatory recruitment alone, we present formal proofs of the supercritical nature of bifurcations (both flip and Hopf) as well as an extensive analysis of dynamics in unstable parameter regions. One important finding here is that in case of small and moderate year to year survival probabilities there are large regions in parameter space where the qualitative behaviour found in a general n+1 dimensional model is retained already in a one-dimensional model. Another result is that the dynamics at or near the boundary of parameter space may be very complicated. Generally, generation delay is found to act as a destabilizing effect but its effect on dynamics is by no means unique. The most profound effect occurs in the n-generation delay cases. In these cases there is no stable equilibrium X ^* at all, but whenever X ^* small, a stable cycle of period n+1 where the periodic points in the cycle are on a very special form. In other cases generation delay does not alter the dynamics in any substantial way. Received 25 April 1995; received in revised form 21 November 1995     

Asymptotic behaviour of a model of hierarchically structured population dynamics

 A hierarchically structured population model with a dependence of the vital rates on a function of the population density (environment) is considered. The existence, uniqueness and the asymptotic behaviour of the solutions is obtained transforming the original non-local PDE of the model into a local one. Under natural conditions, the global asymptotical stability of a nontrivial equilibrium is proved. Finally, if the environment is a function of the biomass distribution, the existence of a positive total biomass equilibrium without a nontrivial population equilibrium is shown. Received 16 February 1996; received in revised form 16 September 1996     

A simple genetic model with non-equilibrium dynamics

 It is implicit in earlier work that simple population genetic models with constant fertility selection at one locus with two alleles can have non-equilibrium dynamics. But the nature of these dynamics has never been investigated in detail. We show that locally stable 2-cycles occur in these models, which seems to be the simplest genetic models exhibiting such dynamics. Received: 5 December 1996 / Revised version: 14 November 1997     

Large time behavior in a nonlinear age-dependent population dynamics problem with spatial diffusion

In this work we analyze the large time behavior in a nonlinear model of population dynamics with age-dependence and spatial diffusion. We show that when t+ either the solution of our problem goes to 0 or it stabilizes to a nontrivial stationary solution. We give two typical examples where the stationary solutions can be evaluated upon solving very simple partial differential equations. As a by-product of the extinction case we find a necessary condition for a nontrivial periodic solution to exist. Numerical computations not described below show a rapid stabilization.This work was partially supported by the Centre National de la Recherche Scientifique through ATP 95939900     

Local dynamics and dispersal in a structured population of the whirligig beetle Deneutus assimilis

The study illustrates the ecological determinants and evolutionary consequences of dispersal in the pond-living water beetle Dineutus assimilis (Coleoptera: Gyrinidae). Over 2 years, local populatiopn dynamics were studied in 51 ponds within a 60-km² study area. In most of the 31 occupied ponds, and even in large populations, abundances changed dramatically from one year to the next. Nine extinction and nine colonisation events were observed. These temporal patterns show no sign of spatial autocorrelation. Such a habitat distribution should favour high dispersal rates. Indeed, D. assimilis was found to be a very effective coloniser of newly available sites (mean propagule size: 23). A mark-recapture study showed that most dispersal occurred after diapause and over distances ranging from 100 m to at least 20 km. Yet despite frequent movement, the local variability in environmental conditions maintiins a large variance in average reproductive success per pond. Furthermore, immigration rates vary widely within a season. The apparent lack of correlation between these two sources of variation should greatly strengthen the role of drift in this system. A companion paper (Nürnberger and Harrison 1995) documents a non-random distribution of mitochondrial haplotypes due to recent population bottlenecks.     

On the formulation and analysis of general deterministic structured population models I. Linear Theory

—We define a linear physiologically structured population model by two rules, one for reproduction and one for “movement” and survival. We use these ingredients to give a constructive definition of next-population-state operators. For the autonomous case we define the basic reproduction ratio R ₀ and the Malthusian parameter r and we compute the resolvent in terms of the Laplace transform of the ingredients. A key feature of our approach is that unbounded operators are avoided throughout. This will facilitate the treatment of nonlinear models as a next step. Received 26 July 1996; received in revised form 3 September 1997     

Chaos and population disappearances in simple ecological models

A class of truncated unimodal discrete-time single species models for which low or high densities result in extinction in the following generation are considered. A classification of the dynamics of these maps into five types is proven: (i) extinction in finite time for all initial densities, (ii) semistability in which all orbits tend toward the origin or a semi-stable fixed point, (iii) bistability for which the origin and an interval bounded away from the origin are attracting, (iv) chaotic semistability in which there is an interval of chaotic dynamics whose compliment lies in the origin’s basin of attraction and (v) essential extinction in which almost every (but not every) initial population density leads to extinction in finite time. Applying these results to the Logistic, Ricker and generalized Beverton-Holt maps with constant harvesting rates, two birfurcations are shown to lead to sudden population disappearances: a saddle node bifurcation corresponding to a transition from bistability to extinction and a chaotic blue sky catastrophe corresponding to a transition from bistability to essential extinction. Received: 14 February 2000 / Revised version: 15 August 2000 / Published online: 16 February 2001     

Evolutionary dynamics of predator-prey systems: an ecological perspective

 Evolution takes place in an ecological setting that typically involves interactions with other organisms. To describe such evolution, a structure is needed which incorporates the simultaneous evolution of interacting species. Here a formal framework for this purpose is suggested, extending from the microscopic interactions between individuals – the immediate cause of natural selection, through the mesoscopic population dynamics responsible for driving the replacement of one mutant phenotype by another, to the macroscopic process of phenotypic evolution arising from many such substitutions. The process of coevolution that results from this is illustrated in the context of predator–prey systems. With no more than qualitative information about the evolutionary dynamics, some basic properties of predator–prey coevolution become evident. More detailed understanding requires specification of an evolutionary dynamic; two models for this purpose are outlined, one from our own research on a stochastic process of mutation and selection and the other from quantitative genetics. Much of the interest in coevolution has been to characterize the properties of fixed points at which there is no further phenotypic evolution. Stability analysis of the fixed points of evolutionary dynamical systems is reviewed and leads to conclusions about the asymptotic states of evolution rather different from those of game-theoretic methods. These differences become especially important when evolution involves more than one species. Received 10 November 1993; received in revised form 25 July 1994     

The diffusion model for migration and selection in a plant population

 The diffusion approximation is derived for migration and selection at a multiallelic locus in a partially selfing plant population subdivided into a lattice of colonies. Generations are discrete and nonoverlapping; both pollen and seeds disperse. In the diffusion limit, the genotypic frequencies at each point are those determined at equilibrium by the local rate of selfing and allelic frequencies. If the drift and diffusion coefficients are taken as the appropriate linear combination of the corresponding coefficients for pollen and seeds, then the migration terms in the partial differential equation for the allelic frequencies have the standard form for a monoecious animal population. The selection term describes selection on the local genotypic frequencies. The boundary conditions and the unidimensional transition conditions for a geographical barrier and for coincident discontinuities in the carrying capacity and migration rate have the standard form. In the diallelic case, reparametrization renders the entire theory of clines and of the wave of advance of favorable alleles directly applicable to plant populations. Received 30 August 1995; received in revised form 23 February 1996     

Global stability of stationary solutions to a nonlinear diffusion equation in phytoplankton dynamics

We consider a nonlinear diffusion equation proposed by Shigesada and Okubo which describes phytoplankton growth dynamics with a selfs-hading effect.We show that the following alternative holds: Either (i) the trivial stationary solution which vanishes everywhere is a unique stationary solution and is globally stable, or (ii) the trivial solution is unstable and there exists a unique positive stationary solution which is globally stable. A criterion for the existence of positive stationary solutions is stated in terms of three parameters included in the equation.     

Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations

In this paper we use a dynamical systems approach to prove the existence of a unique critical value c ^* of the speed c for which the degenerate density-dependent diffusion equation u _ct = [D(u)u _x ] _x + g(u) has: 1. no travelling wave solutions for 0 < c < c ^*, 2. a travelling wave solution u(x, t) = (x - c ^* t) of sharp type satisfying (– ) = 1, () = 0 ^*; '(^*–) = – c ^*/D'(0), '(^*+) = 0 and 3. a continuum of travelling wave solutions of monotone decreasing front type for each c > c ^*. These fronts satisfy the boundary conditions (– ) = 1, '(– ) = (+ ) = '(+ ) = 0. We illustrate our analytical results with some numerical solutions.     

Closing the larval loop: linking larval ecology to the population dynamics of marine benthic invertebrates

The majority of marine benthic invertebrates exhibit a complex life cycle that includes separate planktonic larval, and bottom-dwelling juvenile and adult phases. To understand and predict changes in the spatial and temporal distributions, abundances, population growth rate, and population structure of a species with such a complex life cycle, it is necessary to understand the relative importance of the physical, chemical and biological properties and processes that affect individuals within both the planktonic and benthic phases. To accomplish this goal, it is necessary to study both phases within a common, quantitative framework defined in terms of some common currency. This can be done efficiently through construction and evaluation of a population dynamics model that describes the complete life cycle.

Two forms that such a model might assume are reviewed: a stage-based, population matrix model, and a model that specifies discrete stages of the population, on the bottom and in the water column, in terms of simultaneous differential equations that may be solved in both space and time. Terms to be incorporated in each type of model can be formulated to describe the critical properties and processes that can affect populations within each stage of the life cycle. For both types of model it is shown how this might be accomplished using an idealized balanomorph barnacle as an example species. The critical properties and processes that affect the planktonic and benthic phases are reviewed. For larvae, these include benthic adult fecundity and fertilization success, growth and larval stage duration, mortality, larval behavior, dispersal by currents and turbulence, and larval settlement. It is possible to predict or estimate empirically all of the key terms that should be built into the larval and benthic components of the model. Thus, the challenge of formulating and evaluating a full life cycle model is achievable. Development and evaluation of such a model will be challenging because of the diverse processes which must be considered, and because of the disparities in the spatial and temporal scales appropriate to the benthic and planktonic larval phases. In evaluating model predictions it is critical that sampling schemes be matched to the spatial and temporal scales of model resolution.     

Modelling the depletion and conservation of forestry resources: effects of population and pollution

 In this paper, a mathematical model is proposed to study the depletion of resources in a forest habitat due to the increase of both population and pollution. It is shown that if the rate of pollutant emission into the environment is either population dependent, constant, or periodic, the equilibrium biomass density of the resource settles down to a lower equilibrium than its original carrying capacity, the magnitude of which decreases as the equilibrium levels of the density of population and the concentration of pollutant increase. However, in the case of an instantaneous spill of pollutant into the environment, the equilibrium biomass density decreases with the increase of the equilibrium density of population only. It is found that if the population density and the emission rate of pollutant increase without control, the forestry resource may become extinct. A conservation model is also proposed, the analysis of which shows that the resource biomass can be maintained at a desired level by conserving the forestry resource and by controlling the growth of population and the emission rate of pollutant in the habitat. Received 1 June 1993; received in revised form 1 January 1997     

Correlated extinctions, colonizations and population fluctuations in a highly connected ringlet butterfly metapopulation

 The persistence of metapopulations is likely to be highly dependent on whether population dynamics are correlated among habitat patches as a result of migration between patches and spatially-correlated environmental stochasticity (weather effects). We examined whether population dynamics of the ringlet butterfly, Aphantopus hyperantus, were synchronous in an area of approximately 0.5 km², with respect to extinction, colonization and population fluctuations. Monks Wood Butterfly Monitoring Scheme transect count data from 1973 to 1995, revealed (A) a major environmental perturbation, the drought of 1976, which caused synchronized extinctions of A. hyperantus in subsequent years, (B) synchronized recolonization in years following the large number of apparent extinctions, and (C) population changes by A. hyperantus were highly correlated in many of the 14 sections of the transect, presumably reflecting similar responses to environmental stochasticity, and the exchange of individuals among sections. However, extinction and population synchrony depended on habitat type. Following the 1976 drought, A. hyperantus apparently became extinct from the most open and most shady habitats it occupied, with some persistence in habitats of intermediate shading, thus showing retraction to core populations in central parts of an environmental gradient, albeit with an average shift to relatively open habitat. Populations at extreme ends of the environmental gradient occupied by A. hyperantus fluctuated least synchronously, suggesting a potential buffering effect of habitat heterogeneity, but this was not crucial to survival after the 1976 drought. Thus, not all habitats are equally important to persistence. Correlated temporal dynamics, variation in habitat quality and the interaction between habitat quality and temporal environmental stochasticity are important determinants of metapopulation persistence and should be incorporated in metapopulation models. Received: 26 April 1996 / Accepted: 17 July 1996     

Global stability in consumer-resource cascades

 Models of population growth in consumer-resource cascades (serially arranged containers with a dynamic consumer population, v, receiving a flow of resource, u, from the previous container) with a functional response of the form h(u/v ^b ) are investigated. For b∈[0, 1], it is shown that these models have a globally stable equilibrium. As a result, two conclusions can be drawn: (1) Consumer density dependence in the functional or in the per-capita numerical response can result in persistence of the consumer population in all containers. (2) In the absence of consumer density dependence, the consumer goes extinct in all containers except possibly the first. Several variations of this model are discussed including replacing discrete containers by a spatial continuum and introducing a dynamic resource. Received 25 February 1995 / received in revised form 27 July 1995     

Genealogy and subpopulation differentiation under various models of population structure

 The structured coalescent is used to calculate some quantities relating to the genealogy of a pair of homologous genes and to the degree of subpopulation differentiation, under a range of models of subdivided populations and assuming the infinite alleles model of neutral mutation. The classical island and stepping-stone models of population structure are considered, as well as two less symmetric models. For each model, we calculate the Laplace transform of the distribution of the coalescence time of a pair of genes from specified locations and the corresponding mean and variance. These results are then used to calculate the values of Wright’s coefficient F _ST , its limit as the mutation rate tends to zero and the limit of its derivative with respect to the mutation rate as the mutation rate tends to zero. From this derivative it is seen that F _ST can depend strongly on the mutation rate, for example in the case of an essentially one-dimensional habitat with many subpopulations where gene flow is restricted to neighbouring subpopulations. Received: 1 October 1997 / Revised version: 15 March 1998     

Unravelling the Turing bifurcation using spatially varying diffusion coefficients

. The Turing bifurcation is the basic bifurcation generating spatial pattern, and lies at the heart of almost all mathematical models for patterning in biology and chemistry. In this paper the authors determine the structure of this bifurcation for two coupled reaction diffusion equations on a two-dimensional square spatial domain when the diffusion coefficients have a small explicit variation in space across the domain. In the case of homogeneous diffusivities, the Turing bifurcation is highly degenerate. Using a two variable perturbation method, the authors show that the small explicit spatial inhomogeneity splits the bifurcation into two separate primary and two separate secondary bifurcations, with all solution branches distinct. This splitting of the bifurcation is more effective than that given by making the domain slightly rectangular, and shows clearly the structure of the Turing bifurcation and the way in which the! var ious solution branches collapse together as the spatial variation is reduced. The authors determine the stability of the solution branches, which indicates that several new phenomena are introduced by the spatial variation, including stable subcritical striped patterns, and the possibility that stable stripes lose stability supercritically to give stable spotted patterns.. Received: 10 January 1996/Revised version: 3 July 1996     

Stochastic density dependence in population size of a benthic clonal invertebrate: the regulating role of fission

The influence of environmental variation on the demography of clonal organisms has been poorly studied. I utilise a matrix model of the population dynamics of the intertidal zoanthid Palythoa caesia to examine how density dependence and temporal variation in demographic rates interact in regulating population size. The model produces realistic simulations of population size, with erratic fluctuations between soft lower and upper boundaries of approximately 55 and 90% cover. Cover never exceeds the maximum possible of 100%, and the population never goes to extinction. A sensitivity analysis indicates that the model’s behaviour is driven by density dependence in the fission of large colonies to produce intermediate sized colonies. Importantly, there is no density-dependent mortality in the model, and density dependence in recruitment, while present, is unimportant. Thus it appears that the main demographic processes which are considered to regulate population size in aclonal organisms may not be important for clonal species. Received: 18 August 1999 / Accepted: 29 October 1999     

Convergence in almost periodic Fisher and Kolmogorov models

 We study convergence of positive solutions for almost periodic reaction diffusion equations of Fisher or Kolmogorov type. It is proved that under suitable conditions every positive solution is asymptotically almost periodic. Moreover, all positive almost periodic solutions are harmonic and uniformly stable, and if one of them is spatially homogeneous, then so are others. The existence of an almost periodic global attractor is also discussed. Received: 11 November 1996 / Revised version: 8 January 1998