Patchiness in a minimal nutrient — phytoplankton model

We present a minimal two-component model that can exhibit various types of spatial patterns including patchiness. The model, comprising nutrients and phytoplankton, includes the effect of nutrient uptake by phytoplankton as a Holling type II functional response, and also includes the effect of zooplankton grazing on phytoplankton as a Holling type II non-dynamical term. The mean-field model without the diffusion and advection terms shows both bistability and limit-cycle oscillations as a few parameters such as the input rate of nutrients and the maximum feeding rate of zooplankton are changed. If the parameter values are chosen from the limit-cycle oscillation region, the corresponding reaction-advection-diffusion equations show spatial pattern formations by the combined effects of advection and diffusion by turbulent stirring and mixing, and biological interactions. As the nutrient input is increased, the system behaviour changes from the extinction of the entire phytoplankton to the formation of filamentous patterns, patchiness patterns and homogeneous distributions. These observations suggest that the spatial pattern of phytoplankton can function as an indicator to evaluate the eutrophication level in aquatic ecosystems.     

Dynamics of variable-yield nutrient–phytoplankton–zooplankton models with nutrient recycling and self-shading

 Several nutrient–phytoplankton–zooplankton models with internal nutrient storage by phytoplankton are derived and analyzed. It is shown that there are thresholds beyond which the system is uniformly persistent. Variable-yield models with self-shading of phytoplankton are also considered. With respect to uniform persistence, our result demonstrates that the global dynamics of the system with shading are the same as those for which the self-shading mechanism is ignored. Received: 16 March 1999     

Dynamics of the ‘echo’ effect in a phytoplankton system with nitrogen fixation

Consideration of nitrogen fixation adds a positive nonlinear feedback to plankton ecosystem models. We investigate the consequences of this feedback for secondary phytoplankton blooms and the response of phytoplankton dynamics to physical forcing. The dynamics of phytoplankton, Trichodesmium (the nitrogen fixer), and nutrients is modeled with a system of three differential equations. The model includes two types of nonlinear interactions: the competition of phytoplankton and Trichodesmium for light, and the positive feedback resulting from Trichodesmium recycling. A typical simulation of the model in time, with forcing by a varying mixed-layer depth, reveals a clear successional sequence including a secondary or ‘echo’ bloom of the phytoplankton. We explain this sequence of events through the stability analysis of three different steady states of the model. Our analysis shows the existence of a critical biological parameter, the ratio of normalized growth rates, determining the occurrence of ‘echo’ blooms and the specific sequence of events following a physical perturbation. The interplay of positive and negative feedbacks appears essential to the timing and the type of events following such a perturbation.     

Time delay can enhance spatio-temporal chaos in a prey–predator model

In this paper we explore how the time delay induced Hopf-bifurcation interacts with Turing instability to determine the resulting spatial patterns. For this study, we consider a delayed prey–predator model with Holling type-II functional response and intra-specific competition among the predators. Analytical criteria for the delay induced Hopf-bifurcation and for the delayed spatio-temporal model are provided with numerical example to validate the analytical results. Exhaustive numerical simulation reveals the appearance of three types of stationary patterns, cold spot, labyrinthine, mixture of stripe-spot and two non-stationary patterns, quasi-periodic and spatio-temporal chaotic patterns. The qualitative features of the patterns for the non-delayed and the delayed spatio-temporal model are the same but their occurrence is solely controlled by the temporal parameters, rate of diffusivity and magnitude of the time delay. It is evident that the magnitude of time delay parameter beyond the Hopf-bifurcation threshold mostly produces spatio-temporal chaotic patterns.     

A reaction–diffusion malaria model with incubation period in the vector population

Malaria is one of the most important parasitic infections in humans and more than two billion people are at risk every year. To understand how the spatial heterogeneity and extrinsic incubation period (EIP) of the parasite within the mosquito affect the dynamics of malaria epidemiology, we propose a nonlocal and time-delayed reaction–diffusion model. We then define the basic reproduction ratio R₀{\mathcal{R}_0} and show that R₀{\mathcal{R}_0} serves as a threshold parameter that predicts whether malaria will spread. Furthermore, a sufficient condition is obtained to guarantee that the disease will stabilize at a positive steady state eventually in the case where all the parameters are spatially independent. Numerically, we show that the use of the spatially averaged system may highly underestimate the malaria risk. The spatially heterogeneous framework in this paper can be used to design the spatial allocation of control resources.     

Turing patterns in a predator–prey model with seasonality

Many ecological systems show striking non-homogeneous population distributions. Diffusion-driven instabilities are commonly studied as mechanisms of pattern formation in many fields of biology but only rarely in ecology, in part because some of the conditions seem quite restrictive for ecological systems. Seasonal variation is ubiquitous in temperate ecosystems, yet its effect on pattern formation has not yet been explored. We formulate and analyze an impulsive reaction–diffusion system for a resource and its consumer in a two-season environment. While the resource grows throughout the ‘summer’ season, the consumer reproduces only once per year. We derive conditions for diffusion-driven instability in the system, and we show that pattern formation is possible with a Beddington–DeAngelis functional response. More importantly, we find that a low overwinter survival probability for the resource enhances the propensity for pattern formation: diffusion-driven instability occurs even when the diffusion rates of prey and predator are comparable (although not when they are equal).

     

Evolution of conditional dispersal: a reaction–diffusion–advection model

To study evolution of conditional dispersal, a Lotka-Volterra reaction-diffusion-advection model for two competing species in a heterogeneous environment is proposed and investigated. The two species are assumed to be identical except their dispersal strategies: both species disperse by random diffusion and advection along environmental gradients, but one species has stronger biased movement (i.e., advection along the environmental gradients) than the other one. It is shown that at least two scenarios can occur: if only one species has a strong tendency to move upward the environmental gradients, the two species can coexist since one species mainly pursues resources at places of locally most favorable environments while the other relies on resources from other parts of the habitat; if both species have such strong biased movements, it can lead to overcrowding of the whole population at places of locally most favorable environments, which causes the extinction of the species with stronger biased movement. These results provide a new mechanism for the coexistence of competing species, and they also imply that selection is against excessive advection along environmental gradients, and an intermediate biased movement rate may evolve.     

Boundary forcing-induced oscillations in a phytoplankton–zooplankton patch

We consider one- and two-dimensional minimal models of a phytoplankton–zooplankton dynamical system subject to boundary forced oscillations in the prey. As a consequence, waves propagate in the space dimensions in both species. The difference between the peaks and the troughs are at times of orders of magnitude.     

Persistence and extinction of population in reaction–diffusion–advection model with strong Allee effect growth

Journal of Mathematical Biology - A reaction–diffusion–advection equation with strong Allee effect growth rate is proposed to model a single species stream population in a...     

A stage-structured predator–prey model with distributed maturation delay and harvesting

ABSTRACT

A stage-structured predator–prey system with distributed maturation delay and harvesting is investigated. General birth and death functions are used. The local stability of each feasible equilibria is discussed. By using the persistence theory, it is proven that the system is permanent if the coexistence equilibrium exists. By using Lyapunov functional and LaSalle invariant principle, it is shown that the trivial equilibrium is globally stable when the other equilibria are not feasible, and that the boundary equilibrium is globally stable if the coexistence equilibrium does not exist. Finally, sufficient conditions are derived for the global stability of the coexistence equilibrium.     

Memantine attenuates cognitive impairments after status epilepticus induced in a lithium–pilocarpine model

The capability of memantine, a noncompetitive antagonist of the NMDA receptors, to prevent impairments of cognitive functions in rats was investigated in the lithium–pilocarpine model of epilepsy. After status epilepticus, rats exhibited impaired exploratory behavior and spatial memory, and a decline of extinction of orienting behavior. Memantine administration prevented these disturbances. Thus, the blockade of the NMDA receptors immediately after status epilepticus allowed prevention of the development of the possible cognitive impairments.     

An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects

We present a framework for modeling gliomas growth and their mechanical impact on the surrounding brain tissue (the so-called, mass-effect). We employ an Eulerian continuum approach that results in a strongly coupled system of nonlinear Partial Differential Equations (PDEs): a reaction-diffusion model for the tumor growth and a piecewise linearly elastic material for the background tissue. To estimate unknown model parameters and enable patient-specific simulations we formulate and solve a PDE-constrained optimization problem. Our two main goals are the following: (1) to improve the deformable registration from images of brain tumor patients to a common stereotactic space, thereby assisting in the construction of statistical anatomical atlases; and (2) to develop predictive capabilities for glioma growth, after the model parameters are estimated for a given patient. To our knowledge, this is the first attempt in the literature to introduce an adjoint-based, PDE-constrained optimization formulation in the context of image-driven modeling spatio-temporal tumor evolution. In this paper, we present the formulation, and the solution method and we conduct 1D numerical experiments for preliminary evaluation of the overall formulation/methodology.     

Optimal control of vaccination in a vector-borne reaction–diffusion model applied to Zika virus

Journal of Mathematical Biology - Zika virus has acquired worldwide concern after a recent outbreak in Latin America that started in Brazil, with associated neurological conditions such as...     

Spatiotemporal patterns provoked by environmental variability in a predator–prey model

The emergence of spatiotemporal patterns in the distribution of species is one of the most striking phenomena in ecology and nonlinear science. Since it is known that spatial inhomogeneities can significantly affect the dynamics of ecological populations, in the present paper we investigate the impact of environmental variability on the formation of patterns in a spatially extended predator–prey model. In particular, we utilize a predator–prey system with a Holling III functional response and introduce random spatial variations of the kinetic parameter signifying the intrinsic growth rate of the prey, reflecting the impact of a heterogeneous environment. Our results reveal that in the proximity of the Hopf bifurcation environmental variability is able to provoke pattern formation, whereby the coherence of the patterns exhibits a resonance-like dependence on the variability strength. Furthermore, we show that the phenomenon can only be observed if the spatial heterogeneities exhibit large enough regions with high growth rates of the prey. Our findings thus indicate that variability could be an essential pattern formation mechanism of the populations.     

A general reaction–diffusion model of acidity in cancer invasion

We model the metabolism and behaviour of a developing cancer tumour in the context of its microenvironment, with the aim of elucidating the consequences of altered energy metabolism. Of particular interest is the Warburg Effect, a widespread preference in tumours for cytosolic glycolysis rather than oxidative phosphorylation for glucose breakdown, as yet incompletely understood. We examine a candidate explanation for the prevalence of the Warburg Effect in tumours, the acid-mediated invasion hypothesis, by generalising a canonical non-linear reaction–diffusion model of acid-mediated tumour invasion to consider additional biological features of potential importance. We apply both numerical methods and a non-standard asymptotic analysis in a travelling wave framework to obtain an explicit understanding of the range of tumour behaviours produced by the model and how fundamental parameters govern the speed and shape of invading tumour waves. Comparison with conclusions drawn under the original system—a special case of our generalised system—allows us to comment on the structural stability and predictive power of the modelling framework.     

Reaction–diffusion model of atherosclerosis development

Atherosclerosis begins as an inflammation in blood vessel walls (intima). The inflammatory response of the organism leads to the recruitment of monocytes. Trapped in the intima, they differentiate into macrophages and foam cells leading to the production of inflammatory cytokines and further recruitment of white blood cells. This self-accelerating process, strongly influenced by low-density lipoproteins (cholesterol), results in a dramatic increase of the width of blood vessel walls, formation of an atherosclerotic plaque and, possibly, of its rupture. We suggest a 2D mathematical model of the initiation and development of atherosclerosis which takes into account the concentration of blood cells inside the intima and of pro- and anti-inflammatory cytokines. The model represents a reaction-diffusion system in a strip with nonlinear boundary conditions which describe the recruitment of monocytes as a function of the concentration of inflammatory cytokines. We prove the existence of travelling waves described by this system and confirm our previous results which suggest that atherosclerosis develops as a reaction-diffusion wave. The theoretical results are confirmed by the results of numerical simulations.     

Mathematical analysis of a delayed stage-structured predator–prey model with impulsive diffusion between two predators territories

In most models of population dynamics, diffusion between two patches is assumed to be either continuous or discrete, but in reality, many species diffuse only during a single period, and diffusion often occurs in regular pulses. Further, in forest habitats, the highest-level predator species are restricted to a specific territory, but prey can impulsively move between territories. Therefore, in this paper, we consider a delayed stage-structured predator–prey model with impulsively diffusive prey between two patches; in the model, patches represent the territories of two different predator populations. Here, we analytically obtain the global attractivity condition of predator-extinction periodic solutions for the system by using the concepts of Hui and Chen (2005); a numerical simulation is also included to illustrate this result. Further, we establish permanence conditions for the coexistence of the species using the theory of impulsive delayed differential equations. Finally, we explore the possibilities of the permanence of the system by using the growth rates of immature predators and the impulse period as critical parameters, and we also obtain the parameters’ threshold limits using numerical experimentation.     

Dynamic coexistence in a three-species competition–diffusion system

From the viewpoint of competitor-mediated coexistence, it is important to study the influence of an exotic species on other native species in terms of exogenous effects, in general, exotic species are weaker than native ones because they have evolved in a different environment. Even if an exotic species is weaker, however, it might cooperate with a native species, after which the competitive relations among native species may have reversed. This motivates us to consider the ecological situation whereby one exotic competing species invades the native system of two strongly competing species. Therefore, we discuss the problem of competitive exclusion or competitor-mediated coexistence using a three-species competition–diffusion system.     

A multicomponent reaction–diffusion model of a heterogeneously distributed immobilized enzyme

A physical model was derived for the synthesis of the antibiotic cephalexin with an industrial immobilized penicillin G acylase, called Assemblase. In reactions catalyzed by Assemblase, less product and more by-product are formed in comparison with a free-enzyme catalyzed reaction. The model incorporates reaction with a heterogeneous enzyme distribution, electrostatically coupled transport, and pH-dependent dissociation behavior of reactants and is used to obtain insight in the complex interplay between these individual processes leading to the suboptimal conversion. The model was successfully validated with synthesis experiments for conditions ranging from heavily diffusion limited to hardly diffusion limited, including substrate concentrations from 50 to 600 mM, temperatures between 273 and 303 K, and pH values between 6 and 9. During the conversion of the substrates into cephalexin, severe pH gradients inside the biocatalytic particle, which were previously measured by others, were predicted. Physical insight in such intraparticle process dynamics may give important clues for future biocatalyst design. The modular construction of the model may also facilitate its use for other bioconversions with other biocatalysts.     

Fusing spatial resource heterogeneity with a competition–colonization trade-off in model communities

Two commonly cited mechanisms of multispecies coexistence in patchy environments are spatial heterogeneity in competitive abilities caused by variation in resources and a competition–colonization trade-off. In this paper, a model that fuses these mechanisms together is presented and analyzed. The model suggests that spatial variation in resource ratios can lead to multispecies coexistence, but this mechanism by itself is weak when the number of resources for which species compete is small. However, spatial resource heterogeneity is a powerful mechanism for multispecies coexistence when it acts synergistically with a competition–colonization trade-off. The model also shows how resource supply can control the competitive balance between species that are weak competitors but superior colonizers and strong competitors/inferior colonizers. This provides additional theoretical support for a possible explanation of empirically observed hump-shaped relationships between species diversity and ecological productivity.