We present a novel mathematical model that incorporates role reversal in predator-prey interactions. In particular, we consider the case where adult preys attack and kill juvenile predators. The model is derived from the McKendrick-Von Foerster equation and includes a maturation delay for juvenile predators. We prove that the initial value problem of the modeling framework is globally wellposed and establish conditions for the existence of positive equilibrium points. The model is compared with a version without role reversal, and we use the implicit function theorem to analyze the impact of weak role reversal on the equilibrium points of the model. This approach is necessitated as finding explicit expressions for the equilibrium points in the model with role reversal is particularly challenging. Phase portraits and bifurcation diagrams are used to illustrate our theoretical findings. Our results suggest that the maturation delay of predators and the handling time are the key factors in the dynamics of the model with role reversal, thus highlighting the importance of considering these phenomena in predator-prey interactions. This study provides a novel theoretical observation that the role reversal mechanism in predator-prey interactions has the potential to prevent cyclic population dynamics. The model outcome has been validated with the help of available data for the forage-piscivore fish trade-offs. Experimental studies are needed to validate model predictions and determine its relevance in real -world ecological systems.

Time delays are ubiquitous in industry and nature, and they significantly affect both transient dynamics and stability properties. Consequently, it is often necessary to identify and account for the delays when, e.g., designing a model-based control strategy. However, identifying delays in differential equations is not straightforward and requires specialized methods. Therefore, we propose an algorithm for identifying distributed delays in delay differential equations (DDEs) that only involves simulation of ordinary differential equations (ODEs). Specifically, we 1) approximate the kernel in the DDEs (also called the memory function) by the probability density function of a mixed Erlang distribution and 2) use the linear chain trick (LCT) to transform the resulting DDEs into ODEs. Finally, the parameters in the kernel approximation are estimated as the solution to a dynamical least-squares problem, and we use a single-shooting approach to approximate this solution. We demonstrate the efficacy of the algorithm using numerical examples that involve the logistic equation and a point reactor kinetics model of a molten salt nuclear fission reactor.

We design a linear chain trick algorithm for dynamical systems for which we have oscillatory time histories in the distributed time delay. We make use of this algorithmic framework to analyse memory effects in disease evolution in a population. The modelling is based on a susceptible-infected-recovered SIR-model and on a susceptible-exposed-infected-recovered SEIR-model through a kernel that dampens the activity based on the recent history of infectious individuals. This corresponds to adaptive behavior in the population or through governmental non-pharmaceutical interventions. We use the linear chain trick to show that such a model may be written in a Markovian way, and we analyze the stability of the system. We find that the adaptive behavior gives rise to either a stable equilibrium point or a stable limit cycle for a close to constant number of susceptibles, i.e. locally in time. We also show that the attack rate for this model is lower than it would be without the dampening, although the adaptive behavior disappears as time goes to infinity and the number of infected goes to zero.

We investigate biomass–herbivore–carnivore (top predator) interactions in terms of a tritrophic dynamical systems model. The harvesting rates of the herbivores and the top predators are described by means of a sigmoidal function of the herbivores density and the top predator density, respectively. The main focus in this study is on the dynamics as a function of the natural mortality and the maximal harvesting rate of the top predators. We identify parameter regimes for which we have non-existence of equilibrium points as well as necessary conditions for the existence of such states of the modelling framework. The system does not possess any finite equilibrium states in the regime of high herbivore mortality. In the regime of a high consumption rate of the herbivores and low mortality rates of the top predator, an asymptotically stable finite equilibrium state exists. For this positive equilibrium to exist the mortality of the top predator should not exceed a certain threshold level. We also detect regimes producing coexistence of equilibrium states and their respective stability properties. In the regime of negligible harvesting of the top predator level, we observe a finite window of the natural top predator mortality rates for which oscillations in the top predator-, the herbivore- and the biomass level take place. The lower and upper bound of this window correspond to two Hopf bifurcation points. We also identify a bifurcation diagram using the top predator harvesting rate as a control variable. Using this diagram we detect several saddle node- and Hopf bifurcation points as well as regimes for which we have coexistence of interior equilibrium states, bistability and relaxation type of oscillations.

We analyze the impacts of pollution on fishery sector using a dynamical system approach. The proposed model presupposes that the economic development causes emissions that either remediate or accumulate in the oceans. The model possesses a block structure where the solutions of the rate equations for the pollutant and the economic activity act as an input for the biomass and effort equation. We also account for distributed delay effects in both the pollution level and the economic activity level in our modeling framework. The weight functions in the delay terms are expressed in terms of exponentially decaying functions, which in turn enable us to convert the modeling framework to a higher-order autonomous dynamical system by means of a linear chain trick. When both the typical delay time for the economic activity and the typical delay time for the pollution level are much smaller than the biomass time scale, the governing system is analyzed by means of the theory for singularly perturbed dynamical systems. Contrary to what is found for population dynamical systems with absolute delays, we readily find that the impact of the distributed time lags is negligible in the long-run dynamics in this time-scale separation regime.Recommendations for Resource Managers Resource managers should be aware of the dynamical interrelations between fisheries and pollution, also those that are time delayed.Time lags between biological, economic, and environmental variables are affecting the time paths of these variables.In the long run, the impact from distributed time lags have a negligible influence on the long-run dynamical evolution.Neither the past history of economic activities nor the past history of pollution growth can affect the fishery dynamics in the long run.More empirical and theoretical research are needed to increase our understanding of the interrelation between fisheries and pollution.

In the last decades great advances have been made in mapping neural circuitry of the brain. This has been facilitated by novel experimental techniques for studies both at the single-cell and systems levels. It still remains, though, to combine all the pieces in the puzzle to a coherent picture of brain function. While single nerve cells (neurons) are fairly well understood, the signal - processing properties of the nerve-cell networks in cortex are still obscure. The growth of experimental data has led to a revival of so - called rate equation models for cell networks in nervous tissue (neural networks). In these models, the probability for firing action potentials, the key information carriers in the brain, is the main dynamical variable. These models assume the form of coupled integral and integro - differential equations, and they describe non-linearand nonlocal interactions between the population of excitatory and inhibitory neurons. In the present talk I will discuss the properties of the continuum limit of a 2 population nonlocal Hopfield type of neuronal network model, with spatial periodic microstructure incorporated in the connectivity strength. The modelling framework is derived by means of a homogenization procedure, where the spatial nonlocalities are dealt with by means of Visintins theorem for 2-scale convergence of convolution integrals. I will discuss existence and stability of stationary localized solutions (bumps) and pattern formation though Turing type of instabilities within the framework of this model.

We study pattern formation in a 2-population homogenized neural field model of the Hopfield type in one spatial dimension with periodic microstructure. The connectivity functions are periodically modulated in both the synaptic footprint and in the spatial scale. It is shown that the nonlocal synaptic interactions promote a finite band width instability. The stability method relies on a sequence of wave-number dependent invariants of $2\times 2$ -stability matrices representing the sequence of Fourier-transformed linearized evolution equations for the perturbation imposed on the homogeneous background. The generic picture of the instability structure consists of a finite set of well-separated gain bands. In the shallow firing rate regime the nonlinear development of the instability is determined by means of the translational invariant model with connectivity kernels replaced with the corresponding period averaged connectivity functions. In the steep firing rate regime the pattern formation process depends sensitively on the spatial localization of the connectivity kernels: For strongly localized kernels this process is determined by the translational invariant model with period averaged connectivity kernels, whereas in the complementary regime of weak and moderate localization requires the homogenized model as a starting point for the analysis. We follow the development of the instability numerically into the nonlinear regime for both steep and shallow firing rate functions when the connectivity kernels are modeled by means of an exponentially decaying function. We also study the pattern forming process numerically as a function of the heterogeneity parameters in four different regimes ranging from the weakly modulated case to the strongly heterogeneous case. For the weakly modulated regime, we observe that stable spatial oscillations are formed in the steep firing rate regime, whereas we get spatiotemporal oscillations in the shallow regime of the firing rate functions.

We model bioeconomic interrelations between a commercial fishery and an aquaculture industry by using a dynamical systems theory approach. The biomass follows a logistic growth where the pollution emerging from aquaculture is accounted for by means of a retardation term. We investigate the existence and stability of the equilibrium states of this model as a function of the growth-retardation parameter and find that a necessary (but not sufficient) condition for stability is low and moderate values of the emission-remediation ratio. Three intervals of the growth-retardation parameter are identified in this regime of the emission-remediation ratio. The regime of low and negligible influence of the pollution on the biomass evolution gives rise to the existence of an asymptotically stable equilibrium state characterized by a finite biomass and a finite effort in the fishery. In the same regime we identify two unstable equilibrium states of which the former one is characterized by no effort in the fishery, whereas the latter one is characterized by no biomass and no effort. When the growth retardation parameter exceeds a certain threshold, the fishery becomes unprofitable and the equilibrium state characterized by no effort in the fishery becomes asymptotically stable. By a further increase in this parameter above a higher threshold value, also the biomass is wiped out and the equilibrium state characterized by no biomass and no effort becomes asymptotically stable.

We model pasture-livestock interactions by means of a predator-prey model, with the biomass vegetation as prey and the herbivores as predators. The harvesting rate is a sigmoidal function of the livestock density. We identify the necessary biological and harvest conditions for different equilibria of this model to exist. The system possesses no interior equilibrium points for the mortality rate exceeding a certain threshold. For the regime of low and moderate values of the mortality rate and a high consumption rate per animal, a unique finite and asymptotically stable state exists. We incorporate the effect of forage resource deterioration over time, causing extra decrease in the herbivore population and in the biomass density. We also include the effect of fluctuations in the availability of fodder by allowing for a seasonal periodic variation in the conversion efficiency. This results in extra oscillations superimposed on the general trends of the unperturbed system. Recommendations for Resource Managers Depending on biological and harvest conditions, the system possesses up to three equilibrium states. Forage resource deterioration over time causes an extra decrease in the herbivore population and in the biomass density. A seasonal periodic variation in the conversion efficiency results in oscillations mainly in the herbivore density superimposed on the general trends of the unperturbed system.

We investigate existence and stability of single bumps in a homogenized 2-population neural field model, when the firing rate functions are given by the Heaviside function. The model is derived by means of the two-scale convergence technique of Nguetseng in the case of periodic microvariation in the connectivity functions. The connectivity functions are periodically modulated in both the synaptic footprint and in the spatial scale. The bump solutions are constructed by using a pinning function technique for the case where the solutions are independent of the local variable. In the weakly modulated case the generic picture consists of two bumps (one narrow and one broad bump) for each admissible set of threshold values for firing. In addition, a new threshold value regime for existence of bumps is detected. Beyond the weakly modulated regime the number of bumps depends sensitively on the degree of heterogeneity. For the latter case we present a configuration consisting of three coexisting bumps. The linear stability of the bumps is studied by means of the spectral properties of a Fredholm integral operator, block diagonalization of this operator and the Fourier decomposition method. In the weakly modulated regime, one of the bumps is unstable for all relative inhibition times, while the other one is stable for small and moderate values of this parameter. The latter bump becomes unstable as the relative inhibition time exceeds a certain threshold. In the case of the three coexisting bumps detected in the regime of finite degree of heterogeneity, we have at least one stable bump (and maximum two stable bumps) for small and moderate values of the relative inhibition time.

We obtain conditions for existence of unique global or maximally extended solutions to generalized neural field equations. We also study continuous dependence of these solutions on the spatiotemporal integration kernel, delay effects, firing rate and prehistory functions.

We investigate the existence and stability of localized stationary solutions (referred to as single bumps) in a homogenized 2-population nonlocal neural field model of the Amari type with a Heaviside firing rate function. Solutions are constructed by using a pinning function technique in the case of independence of the local variable. The linear stability for the bumps is investigated by means of the spectral properties of a Fredholm integral operator, block diagonalization of this operator and the Fourier decomposition method.

We study interactions between fishery and aquaculture using a 3D generalized Lotka-Volterra model, where we assume that the aquaculture production may affect the growth rate in the fish stock and the productivity in harvesting. In addition, input demands from both marine industries may result in effort competition. We identify conditions for the coexistence of a unique equilibrium state inside the first octant of the phase space and equilibrium states on its boundary. Conditions for stability and instability of these states are also given, thus showing the possibility of having bistability. The equilibrium point inside the first octant is stable if the growth impact on fishery from sea farming is below the potential productivity in harvesting. In the complementary case, we have an unstable interior equilibrium, and we may then end up in stable equilibrium states on the boundary, where either the fishery or the aquaculture is wiped out. Recommendations for Resource Managers More empirical and theoretical research is needed to reveal types of interrelations between fisheries and aquaculture, and their importance for long run stability between the sectors. When designing policies for the aquaculture industries, managers should in particular be aware of possible long-term harmful effects from aquaculture to fisheries. Increased areas for sea farming reduce the relative profitability of the fishery, and if the area increases above a certain level, this could wipe out the fishery.

Dose-response models are essential to quantitative microbial risk assessment (QMRA), providing a link between levels of human exposure to pathogens and the probability of negative health outcomes. In drinking water studies, the class of semi-mechanistic models known as single-hit models, such as the exponential and the exact beta-Poisson, has seen widespread use. In this work, an attempt is made to carefully develop the general mathematical single-hit framework while explicitly accounting for variation in (1) host susceptibility and (2) pathogen infectivity. This allows a precise interpretation of the so-called single-hit probability and precise identification of a set of statistical independence assumptions that are sufficient to arrive at single-hit models. Further analysis of the model framework is facilitated by formulating the single-hit models compactly using probability generating and moment generating functions. Among the more practically relevant conclusions drawn are: (1) for any dose distribution, variation in host susceptibility always reduces the single-hit risk compared to a constant host susceptibility (assuming equal mean susceptibilities), (2) the model-consistent representation of complete host immunity is formally demonstrated to be a simple scaling of the response, (3) the model-consistent expression for the total risk from repeated exposures deviates (gives lower risk) from the conventional expression used in applications, and (4) a model-consistent expression for the mean per-exposure dose that produces the correct total risk from repeated exposures is developed.

We study existence and continuous dependence of the solutions to the Hammerstein operator equation under the transition from continuous nonlinearities in the Hammerstein operator to the Heaviside nonlinearity in a vicinity of the solution, corresponding to the discontinuous nonlinearity case. We apply these results to corresponding problems arising in the neural activity modeling.

Spatial and/or temporal clustering of pathogens will invalidate the commonly used assumption of Poisson-distributed pathogen counts (doses) in quantitative microbial risk assessment. In this work, the theoretically predicted effect of spatial clustering in conventional single-hit dose-response models is investigated by employing the stuttering Poisson distribution, a very general family of count distributions that naturally models pathogen clustering and contains the Poisson and negative binomial distributions as special cases. The analysis is facilitated by formulating the dose-response models in terms of probability generating functions. It is shown formally that the theoretical single-hit risk obtained with a stuttering Poisson distribution is lower than that obtained with a Poisson distribution, assuming identical mean doses. A similar result holds for mixed Poisson distributions. Numerical examples indicate that the theoretical single-hit risk is fairly insensitive to moderate clustering, though the effect tends to be more pronounced for low mean doses. Furthermore, using Jensen's inequality, an upper bound on risk is derived that tends to better approximate the exact theoretical single-hit risk for highly overdispersed dose distributions. The bound holds with any dose distribution (characterized by its mean and zero inflation index) and any conditional dose-response model that is concave in the dose variable. Its application is exemplified with published data from Norovirus feeding trials, for which some of the administered doses were prepared from an inoculum of aggregated viruses. The potential implications of clustering for dose-response assessment as well as practical risk characterization are discussed.

We show that point-neuron models with a Heaviside firing rate function can be ill posed. More specifically, the initial-condition-to-solution map might become discontinuous in finite time. Consequently, if finite precision arithmetic is used, then it is virtually impossible to guarantee the accurate numerical solution of such models. If a smooth firing rate function is employed, then standard ODE theory implies that point-neuron models are well posed. Nevertheless, in the steep firing rate regime, the problem may become close to ill posed, and the error amplification, in finite time, can be very large. This observation is illuminated by numerical experiments. We conclude that, if a steep firing rate function is employed, then minor round-off errors can have a devastating effect on simulations, unless proper error-control schemes are used.

We investigate existence and stability of rotationally symmetric bump solutions to a homogenized two-dimensional Amari neural field model with periodic micro-variations built in the connectivity strength and by approximating the firing rate function with unit step function. The effect of these variations is parameterized by means of one single parameter, called the degree of heterogeneity. The bumps solutions are assumed to be independent of the microvariable. We develop a framework for study existence of bumps as a function of the degree of heterogeneity as well as a stability method for the bumps. The former problem is based on the pinning function technique while the latter one uses spectral theory for Hilbert-Schmidt integral operators. We demonstrate numerically these procedures for the case when the connectivity kernel is modeled by means of a Mexican hat function. In this case the generic picture consists of one narrow and one broad bump. The radius of the narrow bumps increases with the heterogeneity. For the broad bumps the radius increases for small and moderate values of the activation threshold while it decreases for large values of this threshold. The stability analysis reveals that the narrow bumps remain unstable while the broad bumps are destabilized when the degree of heterogeneity exceeds a certain critical value. (C) 2015 Elsevier B.V. All rights reserved.