Spatiotemporal dynamics of two generic predator–prey models

We present the analysis of two reaction–diffusion systems modelling predator–prey interactions, where the predator displays the Holling type II functional response, and in the absence of predators, the prey growth is logistic. The local analysis is based on the application of qualitative theory for ordinary differential equations and dynamical systems, while the global well-posedness depends on invariant sets and differential inequalities. The key result is an L ^∞-stability estimate, which depends on a polynomial growth condition for the kinetics. The existence of an a priori L ^p -estimate, uniform in time, for all p≥1, implies L ^∞-uniform bounds, given any nonnegative L ^∞-initial data. The applicability of the L ^∞-estimate to general reaction–diffusion systems is discussed, and how the continuous results can be mimicked in the discrete case, leading to stability estimates for a Galerkin finite-element method with piecewise linear continuous basis functions. In order to verify the biological wave phenomena of solutions, numerical results are presented in two-space dimensions, which have interesting ecological implications as they demonstrate that solutions can be ‘trapped’ in an invariant region of phase space.     

On the stationary state analysis of reaction-diffusion mechanisms for biological pattern formation

We present a biologically plausible two-variable reaction-diffusion model for the developing vertebrate limb, for which we postulate the existence of a stationary solution. A consequence of this assumption is that the stationary state depends on only a single concentration-variable. Under these circumstances, features of potential biological significance, such as the dependence of the steady-state concentration profile of this variable on parameters such as tissue size and shape, can be studied without detailed information about the rate functions. As the existence and stability of stationary solutions, which must be assumed for any biochemical system governing morphogenesis, cannot be investigated without such information, an analysis is made of the minimal requirements for stable, stationary non-uniform solutions in a general class of reaction-diffusion systems. We discuss the strategy of studying stationary-state properties of systems that are incompletely specified. Where abrupt transitions between successive compartment-sizes occur, as in the developing limb, we argue that it is reasonable to model pattern reorganization as a sequence of independent stationary states.     

A reaction-diffusion theory of morphogenesis with inherent pattern invariance under scale variations

In the framework of reaction-diffusion theory we deal with the problem of pattern regulation in morphogenesis. A generic model is proposed where the kinetic terms follow constraints imposed by scale invariance considerations. These constraints allow a class of kinetic schemes to be formulated so that, starting with an initially homogeneous morphogen distribution in the field, a stable gradient is established of the form: S(chi,L) = Lpf(chi/L). Here L is the length of the morphogenetic field, chi is the position variable and f(chi/L) is some monotonic function of the relative distance. With this distribution a scale invariant gradient can be constructed which leads to pattern regulation. A linear stability analysis of the model permits the definition of the parameter values enabling the system to abandon the homogeneous state spontaneously. Simulations of the evolution of the system towards its final stable state result in approximate pattern invariance for different field lengths. The accuracy of this invariance is in agreement with some recent quantitative experimental findings in both developing and regenerating systems.     

Implications of a warming North Sea for the growth of haddock Melanogrammus aeglefinus

The present study aimed firstly, to test for a temperature effect on North Sea haddock Melanogrammus aeglefinus growth and secondly, to develop a model that could be used to assess total length (L(T)) and mass (M)-at-age response to different temperature scenarios. The von Bertalanffy growth model was fitted on a cohort-by-cohort basis from 1970 to 2006. The asymptotic L(T) (L(∞)) was negatively correlated with temperature while the rate at which L(∞) is reached (K) was positively correlated with temperature. K was negatively correlated with density, whereas no effect on L(∞) was observed. These effects were incorporated into a von Bertalanffy model which was extended to include temperature and density as explanatory variables. Only the temperature variable was significant. Fitting the extended von Bertalanffy model revealed that L(∞) decreased while K increased with increasing temperature, resulting in up to a 40% loss of individual yield at older ages. The dramatic decline observed in the mean age at which 50% of the population becomes mature suggests that higher temperatures resulted in larger young M. aeglefinus that matured earlier and therefore reached a smaller maximum size. In a global warming context, the loss of individual yield observed at old ages is likely to reduce the fisheries yield for M. aeglefinus in the North Sea.     

Bifurcation analysis of nonlinear reaction-diffusion equations—II. Steady state solutions and comparison with numerical simulations

The steady state spatial patterns arising in nonlinear reaction-diffusion systems beyond an instability point of the thermodynamic branch are studied on a simple model network. A detailed comparison between the analytical solutions of the kinetic equations, obtained by bifurcation theory, and the results of computer simulations is presented for different boundary conditions. The characteristics of the dissipative structures are discussed and it is shown that the observed behavior depends strongly on both the boundary and initial conditions. The theoretical expressions are limited to the neighborhood of the marginal stability point. Computer simulations allow not only the verification of their predictions but also the investigation of the behavior of the system for larger deviations from the instability point. It is shown that new features such as multiplicity of solutions and secondary bifurcations can appear in this region.     

Robust stabilization and H(∞) controllers design for stochastic genetic regulatory networks with time-varying delays and structured uncertainties

This paper deals with the problem of stabilization design and H(∞) control for a class of genetic regulatory networks (GRNs) with both intrinsic perturbation and extrinsic perturbation. Some delay-dependent mean-square stabilization criteria are put forward for nominal systems and uncertain systems by using an improved free-weighting matrix approach. As a result, the corresponding stabilization controllers and H(∞) controllers of GRNs are constructed with time delays compensated and suboptimal solutions are obtained via exploiting an iterative procedure together with the linear matrix inequality (LMI) method and the cone complementarity liberalization (CCL) algorithm. Finally, three numerical examples are presented to illustrate the effectiveness of the proposed theoretical results.     

The surface finite element method for pattern formation on evolving biological surfaces

In this article we propose models and a numerical method for pattern formation on evolving curved surfaces. We formulate reaction-diffusion equations on evolving surfaces using the material transport formula, surface gradients and diffusive conservation laws. The evolution of the surface is defined by a material surface velocity. The numerical method is based on the evolving surface finite element method. The key idea is based on the approximation of Γ by a triangulated surface Γ _h consisting of a union of triangles with vertices on Γ. A finite element space of functions is then defined by taking the continuous functions on Γ _h which are linear affine on each simplex of the polygonal surface. To demonstrate the capability, flexibility, versatility and generality of our methodology we present results for uniform isotropic growth as well as anisotropic growth of the evolution surfaces and growth coupled to the solution of the reaction-diffusion system. The surface finite element method provides a robust numerical method for solving partial differential systems on continuously evolving domains and surfaces with numerous applications in developmental biology, tumour growth and cell movement and deformation.     

The life histories of endangered hammerhead sharks (Carcharhiniformes, Sphyrnidae) from the east coast of Australia

The life histories of two globally endangered hammerhead sharks, Sphyrna lewini and Sphyrna mokarran, were examined using samples collected from a range of commercial fisheries operating along the east coast of Australia. The catch of S. lewini was heavily biased towards males, and there were significant differences in von Bertalanffy growth parameters (L(∞) and k) and maturity [stretched total length (L(ST)) and age (A) at which 50% are mature, L(ST50) and A(50)] between those caught in the tropics (L(∞) = 2119 mm, k = 0·163, L(ST50) = 1471 mm, A(50) = 5·7 years) and those caught in temperate waters (L(∞) = 3199 mm, k = 0·093, L(ST50) = 2043 mm, A(50) = 8·9 years). The best-fit estimates for a three-parameter von Bertalanffy growth curve fit to both sexes were L(∞) = 3312 mm, L(0) = 584 mm and k = 0·076. Males attained a maximum age of 21 years and grew to at least 2898 mm L(ST). The longevity, maximum length and maturity of females could not be estimated as mature animals could not be sourced from any fishery. Length at birth inferred from neonates with open umbilical scars was 465-563 mm L(ST). There was no significant difference in length and age at maturity of male and female S. mokarran, which reached 50% maturity at 2279 mm L(ST) and 8·3 years. Sphyrna mokarran grew at a similar rate to S. lewini and the best-fit estimates for a two-parameter von Bertalanffy equation fit to length-at-age data for sexes combined with an assumed mean length-at-birth of 700 mm were L(∞) = 4027 mm and k = 0·079. Females attained a maximum age of 39·1 years and grew to at least 4391 mm L(ST). The oldest male S. mokarran was 31·7 years old and 3691 mm L(ST). Validation of annual growth-band deposition in S. mokarran was achieved through a mark, tag and recapture study.     

Dynamic instabilities in the microtubule cytoskeleton: A state diagram

The dynamics of microtubule growth and disassembly is considered in the framework of the theory of nonequilibrium reaction-diffusion systems. The phase diagram contains regions corresponding to stable stationary and nonstationary solutions. Dynamic instabilities can arise from nonequilibrium kinetic transitions. Agents affecting the microtubule dynamics are classed into four types, and the interplay of their effects is analyzed.     

Age and growth of albacore Thunnus alalunga in the North Pacific Ocean

The age and growth of North Pacific albacore Thunnus alalunga were investigated using obliquely sectioned sagittal otoliths from samples of 126 females and 148 males. Otolith edge analysis indicated that the identified annulus in a sagittal otolith is primarily formed during the period from September to February. The assessments of the fish age at first annulus formation indicated that the first annulus represents an age of <1 year. This study presents an age estimate (0·75 years) for the formation of the first annulus. The oldest fish ages observed in this study were 10 years for females and 14 years for males. The von Bertalanffy growth parameters of females estimated were L(∞) = 103·5 cm in fork length (L(F) ), K = 0·340 year(-1) and t(0) = -0·53 years, and the parameters of males were L(∞) = 114·0 cm, K = 0·253 year(-1) and t(0) = -1·01 years. Sexual size dimorphism between males and females seemed to occur after reaching sexual maturity. The coefficients of the power function for expressing the L(F) -mass relationship obtained from sex-pooled data were a = 2·964 × 10(-5) and b = 2·928.     

Effect of environmental fluctuations on invasion fronts

We determine the density profile and velocity of invasion fronts in one-dimensional infinite habitats in the presence of environmental fluctuations. The population dynamics is reformulated in terms of a stochastic reaction-diffusion equation and is reduced to a deterministic equation that incorporates the systematic contributions of the noise. We obtain analytical expressions for the front profile and velocity by constructing a variational principle. The effect of the noise differs, depending on whether it affects the density-independent growth rate, the intraspecific competition term or the Allee threshold. Fluctuations in the density-independent growth rate increase the invasion velocity and the population density of the invaded area. Fluctuations in the competition term also change the population density of the invaded area, but modify the invasion velocity only for certain initial conditions. Fluctuations in the Allee threshold can induce pulled or pushed invasion fronts as well as invasion failure. We compare our analytical results with numerical solutions of the stochastic partial differential equations and show that our procedure proves useful in dealing with reaction-diffusion equations with multiplicative noise.     

Galerkin-Ritz procedures for approximate solutions to systems of reaction-diffusion equations

For any essentially nonlinear system of reaction-diffusion equations of the generic form ∂c_i/∂t=D_i∇²c_i+Q_i(c,x,t) supplemented with Robin type boundary conditions over the surface of a closed bounded three-dimensional region, it is demonstrated that all solutions for the concentration distributionn-tuple function c=(c ₁(x,t),...,c _n (x,t)) satisfy a differential variational condition. Approximate solutions to the reaction-diffusion intial-value boundary-value problem are obtainable by employing this variational condition in conjunction with a Galerkin-Ritz procedure. It is shown that the dynamical evolution from a prescribed initial concentrationn-tuple function to a final steady-state solution can be determined to desired accuracy by such an approximation method. The variational condition also admits a systematic Galerkin-Ritz procedure for obtaining approximate solutions to the multi-equation elliptic boundary-value problem for steady-state distributions c=−c(x). Other systems of phenomenological (non-Lagrangian) field equations can be treated by Galerkin-Ritz procedures based on analogues of the differential variational condition presented here. The method is applied to derive approximate nonconstant steady-state solutions for ann-species symbiosis model.     

Population dynamics and life history of a geographically restricted seahorse, Hippocampus whitei

The aim of this study was to collect data on population dynamics and life history for White's seahorse Hippocampus whitei, a geographically restricted species that is listed as data deficient under the IUCN Red List. Data from H. whitei populations were collected from two regions, Port Stephens (north) and Sydney Harbour (south) in New South Wales, Australia, covering most of the known range of H. whitei, from 2005 to 2010. Over 1000 individuals were tagged using fluorescent elastomer and on subsequent recaptures were re-measured for growth data that were used in a forced Gulland-Holt plot to develop growth parameters for use in a specialized von Bertalanffy growth-function model. Growth parameters for Port Stephens were: females L(∞) = 149·2 mm and K = 2·034 per year and males L(∞) = 147·9 mm and K = 2·520 per year compared with estimates from Sydney Harbour: females L(∞) = 139·8 mm and K = 1·285 per year and males L(∞) = 141·6 mm and K = 1·223 per year. Whilst there was no significant difference in growth between sexes for each region, H. whitei in Port Stephens grew significantly quicker and larger and matured and reproduced at a younger age than those from Sydney Harbour. The life span of H. whitei is at least 5 years in the wild with six individuals recorded reaching this age. Data collected on breeding pairs found that H. whitei displays life-long monogamy with three pairs observed remaining pair bonded over three consecutive breeding years. Baseline population densities were derived for two Port Stephens' sites (0·035 and 0·110 m(-2) ) and for Manly in Sydney Harbour (1·050 m(-2) ). Even though the life-history parameters of H. whitei suggest it may be reasonably resilient, precaution should be taken in its future management as a result of its limited geographical distribution and increasing pressures from anthropogenic sources on its habitats.     

Critical thresholds for eventual extinction in randomly disturbed population growth models

This paper considers several single species growth models featuring a carrying capacity, which are subject to random disturbances that lead to instantaneous population reduction at the disturbance times. This is motivated in part by growing concerns about the impacts of climate change. Our main goal is to understand whether or not the species can persist in the long run. We consider the discrete-time stochastic process obtained by sampling the system immediately after the disturbances, and find various thresholds for several modes of convergence of this discrete process, including thresholds for the absence or existence of a positively supported invariant distribution. These thresholds are given explicitly in terms of the intensity and frequency of the disturbances on the one hand, and the population’s growth characteristics on the other. We also perform a similar threshold analysis for the original continuous-time stochastic process, and obtain a formula that allows us to express the invariant distribution for this continuous-time process in terms of the invariant distribution of the discrete-time process, and vice versa. Examples illustrate that these distributions can differ, and this sends a cautionary message to practitioners who wish to parameterize these and related models using field data. Our analysis relies heavily on a particular feature shared by all the deterministic growth models considered here, namely that their solutions exhibit an exponentially weighted averaging property between a function of the initial condition, and the same function applied to the carrying capacity. This property is due to the fact that these systems can be transformed into affine systems.

     

Diffusion versus network models as descriptions for the spread of prion diseases in the brain

In this paper we will discuss different modeling approaches for the spread of prion diseases in the brain. Firstly, we will compare reaction-diffusion models with models of epidemic diseases on networks. The solutions of the resulting reaction-diffusion equations exhibit traveling wave behavior on a one-dimensional domain, and the wave speed can be estimated. The models can be tested for diffusion-driven (Turing) instability, which could present a possible mechanism for the formation of plaques. We also show that the reaction-diffusion systems are capable of reproducing experimental data on prion spread in the mouse visual system. Secondly, we study classical epidemic models on networks, and use these models to study the influence of the network topology on the disease progression.     

The diffusive spread of alleles in heterogeneous populations

The spread of genes and individuals through space in populations is relevant in many biological contexts. I study, via systems of reaction-diffusion equations, the spatial spread of advantageous alleles through structured populations. The results show that the temporally asymptotic rate of spread of an advantageous allele, a kind of invasion speed, can be approximated for a class of linear partial differential equations via a relatively simple formula, c = 2 square root of (rD), that is reminiscent of a classic formula attributed to R. A. Fisher. The parameters r and D represent an asymptotic growth rate and an average diffusion rate, respectively, and can be interpreted in terms of eigenvalues and eigenvectors that depend on the population's demographic structure. The results can be applied, under certain conditions, to a wide class of nonlinear partial differential equations that are relevant to a variety of ecological and evolutionary scenarios in population biology. I illustrate the approach for computing invasion speed with three examples that allow for heterogeneous dispersal rates among different classes of individuals within model populations.     

一类具连续时滞的三种群互助模型正平衡态的渐近稳定性

本文研究了一类具连续时滞的三种群互助模型,利用上、下解方法及相应的单调迭代方法,获得了该系统存在唯一正常数平衡态及该平衡态是全局渐近稳定的结论,为讨论时滞三种群模型提供了一种有效方法,所得结果也适用于二种群互助模型及不含时滞和扩散项的互助模型,因而推广了已有的一些结论．     

An alternative formulation for a delayed logistic equation

We derive an alternative expression for a delayed logistic equation, assuming that the rate of change of the population depends on three components: growth, death, and intraspecific competition, with the delay in the growth component. In our formulation, we incorporate the delay in the growth term in a manner consistent with the rate of instantaneous decline in the population given by the model. We provide a complete global analysis, showing that, unlike the dynamics of the classical logistic delay differential equation (DDE) model, no sustained oscillations are possible. Just as for the classical logistic ordinary differential equation (ODE) growth model, all solutions approach a globally asymptotically stable equilibrium. However, unlike both the logistic ODE and DDE growth models, the value of this equilibrium depends on all of the parameters, including the delay, and there is a threshold that determines whether the population survives or dies out. In particular, if the delay is too long, the population dies out. When the population survives, i.e., the attracting equilibrium has a positive value, we explore how this value depends on the parameters. When this value is positive, solutions of our DDE model seem to be well approximated by solutions of the logistic ODE growth model with this carrying capacity and an appropriate choice for the intrinsic growth rate that is independent of the initial conditions.