Nonlinear Analysis-real World Applications, Feb 1, 2011
We review and complement the study of the mathematical properties of the predator-prey dynamical ... more We review and complement the study of the mathematical properties of the predator-prey dynamical system for spruce-budworm interactions studied in Ludwig et al. [D. Ludwig, D. Jones, C.S. Holling, Qualitative analysis of insect outbreak systems: the spruce budworm and forest, Journal of Animal Ecology 47 (1978), 315-332]. We use the singular perturbation method to identify parameter regimes which permit relaxation oscillations. The leading order contribution to the period of these oscillations is computed by means of this method and compared with the outcome of direct numerical simulations.
We model bioeconomic interrelations between a commercial fishery and an aquaculture industry by u... more 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 analyze the possible impacts of pollution on a fishery by means of a dynamical systems theory ... more We analyze the possible impacts of pollution on a fishery by means of a dynamical systems theory approach. The proposed model presupposes that activities stimulating economic growth also cause higher emissions that remediate or accumulate in the oceans. The density of pollution is assumed to affect the fishery negatively by reducing biological growth potential and decreasing marginal willingness to pay for the fish in the market. Additionally, economic growth increases the general income and may also increase the demand for fish. We show that the modelling framework permits a unique stable equilibrium state in the regime of moderate values of the emission-remediation ratio. We also investigate how the ecological and market impacts alter both the steady state and dynamics of an open access fishery.
We study interactions between fishery and aquaculture using a 3D generalized Lotka–Volterra model... more 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 empir...
Communications in Nonlinear Science and Numerical Simulation, Mar 1, 2016
We investigate existence and stability of rotationally symmetric bump solutions to a homogenized ... more 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 micro-variable. 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.
We analyze the impacts of pollution on fishery sector using a dynamical system approach. The prop... more 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.
h i g h l i g h t s • Homogenization of a 2-population neural field model. • Existence and unique... more h i g h l i g h t s • Homogenization of a 2-population neural field model. • Existence and uniqueness of single bumps. • Stability of single bumps.
P ro ceed in gs of t h e I n t ern a ti o nal W o rk sho p N O A '02 , Êu k ≤ci n 200 2 O p t ica... more P ro ceed in gs of t h e I n t ern a ti o nal W o rk sho p N O A '02 , Êu k ≤ci n 200 2 O p t ical B eam s in N on lo cal N on lin ear Me d ia W. K r § l ik ow sk i A u st r al i an P hoto ni cs Coop erati ve Research Centre , La ser Ph ysi cs Centre R esearch School of Physi cal Science and En gi neeri ng The Austra l i an Nati onal Uni versi ty , Can b erra ACT 0200, Austra l i a O. B ang Depart m ent o f Ma them ati cal Mo dell i ng T echnical Uni versi ty D enmar k, Bui l di ng 305 / 321, 28 00 Kg s. Lyng by, D enmark J. W y l l er D epartm ent of Ma them ati cal Sciences, Ag ri cul tura l Uni versi ty of Nor way
Journal of Mathematical Analysis and Applications, 2016
We study existence and continuous dependence of the solutions to the Hammerstein operator equatio... more 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.
QMRA for Drinking Water: 2. The Effect of Pathogen Clustering in Single-Hit Dose-Response Models
Risk Analysis, 2016
Spatial and/or temporal clustering of pathogens will invalidate the commonly used assumption of P... more 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.
Nonlinear Analysis-real World Applications, Feb 1, 2011
We review and complement the study of the mathematical properties of the predator-prey dynamical ... more We review and complement the study of the mathematical properties of the predator-prey dynamical system for spruce-budworm interactions studied in Ludwig et al. [D. Ludwig, D. Jones, C.S. Holling, Qualitative analysis of insect outbreak systems: the spruce budworm and forest, Journal of Animal Ecology 47 (1978), 315-332]. We use the singular perturbation method to identify parameter regimes which permit relaxation oscillations. The leading order contribution to the period of these oscillations is computed by means of this method and compared with the outcome of direct numerical simulations.
We model bioeconomic interrelations between a commercial fishery and an aquaculture industry by u... more 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 analyze the possible impacts of pollution on a fishery by means of a dynamical systems theory ... more We analyze the possible impacts of pollution on a fishery by means of a dynamical systems theory approach. The proposed model presupposes that activities stimulating economic growth also cause higher emissions that remediate or accumulate in the oceans. The density of pollution is assumed to affect the fishery negatively by reducing biological growth potential and decreasing marginal willingness to pay for the fish in the market. Additionally, economic growth increases the general income and may also increase the demand for fish. We show that the modelling framework permits a unique stable equilibrium state in the regime of moderate values of the emission-remediation ratio. We also investigate how the ecological and market impacts alter both the steady state and dynamics of an open access fishery.
We study interactions between fishery and aquaculture using a 3D generalized Lotka–Volterra model... more 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 empir...
Communications in Nonlinear Science and Numerical Simulation, Mar 1, 2016
We investigate existence and stability of rotationally symmetric bump solutions to a homogenized ... more 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 micro-variable. 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.
We analyze the impacts of pollution on fishery sector using a dynamical system approach. The prop... more 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.
h i g h l i g h t s • Homogenization of a 2-population neural field model. • Existence and unique... more h i g h l i g h t s • Homogenization of a 2-population neural field model. • Existence and uniqueness of single bumps. • Stability of single bumps.
P ro ceed in gs of t h e I n t ern a ti o nal W o rk sho p N O A '02 , Êu k ≤ci n 200 2 O p t ica... more P ro ceed in gs of t h e I n t ern a ti o nal W o rk sho p N O A '02 , Êu k ≤ci n 200 2 O p t ical B eam s in N on lo cal N on lin ear Me d ia W. K r § l ik ow sk i A u st r al i an P hoto ni cs Coop erati ve Research Centre , La ser Ph ysi cs Centre R esearch School of Physi cal Science and En gi neeri ng The Austra l i an Nati onal Uni versi ty , Can b erra ACT 0200, Austra l i a O. B ang Depart m ent o f Ma them ati cal Mo dell i ng T echnical Uni versi ty D enmar k, Bui l di ng 305 / 321, 28 00 Kg s. Lyng by, D enmark J. W y l l er D epartm ent of Ma them ati cal Sciences, Ag ri cul tura l Uni versi ty of Nor way
Journal of Mathematical Analysis and Applications, 2016
We study existence and continuous dependence of the solutions to the Hammerstein operator equatio... more 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.
QMRA for Drinking Water: 2. The Effect of Pathogen Clustering in Single-Hit Dose-Response Models
Risk Analysis, 2016
Spatial and/or temporal clustering of pathogens will invalidate the commonly used assumption of P... more 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.
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