Spatiotemporal Dynamics of the Epidemic Transmission in a Predator-Prey System

Epidemic transmission is one of the critical density-dependent mechanisms that affect species viability and dynamics. In a predator-prey system, epidemic transmission can strongly affect the success probability of hunting, especially for social animals. Predators, therefore, will suffer from the positive density-dependence, i.e., Allee effect, due to epidemic transmission in the population. The rate of species contacting the epidemic, especially for those endangered or invasive, has largely increased due to the habitat destruction caused by anthropogenic disturbance. Using ordinary differential equations and cellular automata, we here explored the epidemic transmission in a predator-prey system. Results show that a moderate Allee effect will destabilize the dynamics, but it is not true for the extreme Allee effect (weak or strong). The predator-prey dynamics amazingly stabilize by the extreme Allee effect. Predators suffer the most from the epidemic disease at moderate transmission probability. Counter-intuitively, habitat destruction will benefit the control of the epidemic disease. The demographic stochasticity dramatically influences the spatial distribution of the system. The spatial distribution changes from oil-bubble-like (due to local interaction) to aggregated spatially scattered points (due to local interaction and demographic stochasticity). It indicates the possibility of using human disturbance in habitat as a potential epidemic-control method in conservation.     

Impacts of Biotic Resource Enrichment on a Predator–Prey Population

The environmental carrying capacity is usually assumed to be fixed quantity in the classical predator–prey population growth models. However, this assumption is not realistic as the environment generally varies with time. In a bid for greater realism, functional forms of carrying capacities have been widely applied to describe varying environments. Modelling carrying capacity as a state variable serves as another approach to capture the dynamical behavior between population and its environment. The proposed modified predator–prey model is based on the ratio-dependent models that have been utilized in the study of food chains. Using a simple non-linear system, the proposed model can be linked to an intra-guild predation model in which predator and prey share the same resource. Distinct from other models, we formulate the carrying capacity proportional to a biotic resource and both predator and prey species can directly alter the amount of resource available by interacting with it. Bifurcation and numerical analyses are presented to illustrate the system’s dynamical behavior. Taking the enrichment parameter of the resource as the bifurcation parameter, a Hopf bifurcation is found for some parameter ranges, which generate solutions that posses limit cycle behavior.     

Impacts of Incubation Delay on the Dynamics of an Eco-Epidemiological System—A Theoretical Study

Parasite and predator play significant role in trophic interaction, productivity and stability of an ecosystem. In this paper, we have studied a host-parasite-predator interaction that incorporates incubation delay. How the qualitative and quantitative behaviors of the system alter with the incubation delay have been discussed both from mathematical and biological point of views. It is observed that for a lower infection rate, the system is stable for all delays; but for a higher infection rate, there exists a threshold value of the delay above which the system is unstable and below which the system is stable leading to the persistence of all the species. Also, the instability arising from the incubation delay may be controlled if somehow the growth rate of predator population is increased. Numerical studies have also been performed to illustrate different analytical findings. Research is supported by UGC, India; F No. 32-173/2006(SR).     

Dynamics of a Prey–Predator System with Herd Behaviour in Both and Strong Allee Effect in Prey

Biophysics - This paper mainly deals with the prey?predator dynamics where both the prey and predator exhibit herd behavior. Positivity, boundedness, some extinction criteria, stability of...     

Evolutionary Behaviour,Trade-Offs and Cyclic and Chaotic Population Dynamics

Many studies of the evolution of life-history traits assume that the underlying population dynamical attractor is stable point equilibrium. However, evolutionary outcomes can change significantly in different circumstances. We present an analysis based on adaptive dynamics of a discrete-time demographic model involving a trade-off whose shape is also an important determinant of evolutionary behaviour. We derive an explicit expression for the fitness in the cyclic region and consequently present an adaptive dynamic analysis which is algebraic. We do this fully in the region of 2-cycles and (using a symbolic package) almost fully for 4-cycles. Simulations illustrate and verify our results. With equilibrium population dynamics, trade-offs with accelerating costs produce a continuously stable strategy (CSS) whereas trade-offs with decelerating costs produce a non-ES repellor. The transition to 2-cycles produces a discontinuous change: the appearance of an intermediate region in which branching points occur. The size of this region decreases as we move through the region of 2-cycles. There is a further discontinuous fall in the size of the branching region during the transition to 4-cycles. We extend our results numerically and with simulations to higher-period cycles and chaos. Simulations show that chaotic population dynamics can evolve from equilibrium and vice-versa.     

Pattern Formation,Long-Term Transients,and the Turing–Hopf Bifurcation in a Space- and Time-Discrete Predator–Prey System

Understanding of population dynamics in a fragmented habitat is an issue of considerable importance. A natural modelling framework for these systems is spatially discrete. In this paper, we consider a predator–prey system that is discrete both in space and time, and is described by a Coupled Map Lattice (CML). The prey growth is assumed to be affected by a weak Allee effect and the predator dynamics includes intra-specific competition. We first reveal the bifurcation structure of the corresponding non-spatial system. We then obtain the conditions of diffusive instability on the lattice. In order to reveal the properties of the emerging patterns, we perform extensive numerical simulations. We pay a special attention to the system properties in a vicinity of the Turing–Hopf bifurcation, which is widely regarded as a mechanism of pattern formation and spatiotemporal chaos in space-continuous systems. Counter-intuitively, we obtain that the spatial patterns arising in the CML are more typically stationary, even when the local dynamics is oscillatory. We also obtain that, for some parameter values, the system’s dynamics is dominated by long-term transients, so that the asymptotical stationary pattern arises as a sudden transition between two different patterns. Finally, we argue that our findings may have important ecological implications.     

Modeling the Effect of Prey Refuge on a Ratio-Dependent Predator–Prey System with the Allee Effect

The extinction of species is a major threat to the biodiversity. The species exhibiting a strong Allee effect are vulnerable to extinction due to predation. The refuge used by species having a strong Allee effect may affect their predation and hence extinction risk. A mathematical study of such behavioral phenomenon may aid in management of many endangered species. However, a little attention has been paid in this direction. In this paper, we have studied the impact of a constant prey refuge on the dynamics of a ratio-dependent predator–prey system with strong Allee effect in prey growth. The stability analysis of the model has been carried out, and a comprehensive bifurcation analysis is presented. It is found that if prey refuge is less than the Allee threshold, the incorporation of prey refuge increases the threshold values of the predation rate and conversion efficiency at which unconditional extinction occurs. Moreover, if the prey refuge is greater than the Allee threshold, situation of unconditional extinction may not occur. It is found that at a critical value of prey refuge, which is greater than the Allee threshold but less than the carrying capacity of prey population, system undergoes cusp bifurcation and the rich spectrum of dynamics exhibited by the system disappears if the prey refuge is increased further.     

Modeling the Dynamics of Viral Evolution Considering Competition Within Individual Hosts and at Population Level: The Effects of Treatment

We consider two viral strains competing against each other within individual hosts (at cellular level) and at population level (for infecting hosts) by studying two cases. In the first case, the strains do not mutate into each other. In this case, we found that each individual in the population can be infected by only one strain and that co-existence in the population is possible only when the strain that has the greater basic intracellular reproduction number, R _0c , has the smaller population number R _0p . Treatment against the one strain shifts the population equilibrium toward the other strain in a complicated way (see Appendix B). In the second case, we assume that the strain that has the greater intracellular number R _0c can mutate into the other strain. In this case, individual hosts can be simultaneously infected by both strains (co-existence within the host). Treatment shifts the prevalence of the two strains within the hosts, depending on the mortality induced by the treatment, which is, in turn, dependent upon the doses given to each individual. The relative proportions of the strains at the population level, under treatment, depend both on the relative proportions within the hosts (which is determined by the dosage of treatment) and on the number of individuals treated per unit time, that is, the rate of treatment. Implications for cases of real diseases are briefly discussed.     

Attractor Dynamics and Thermodynamic Analogies in the Cerebral Cortex: Synchronous Oscillation,the Background EEG,and the Regulation of Attention

Ongoing changes in attention and cognition depend upon cortical/subcortical interactions, which select sequences of different spatial patterns of activation in the cortex.It is proposed that each pattern of cortical activation permits evolution of electrocortical wave activity toward statistically stationary states, analogous to thermodynamic equilibrium. In each steady-state, neurons fire with an intrinsic Poisson spike probability and also with a bursting pattern related to network oscillations. Excitatory cell dendrites act as a regenerative reservoir in which pulse generation is balanced against dissipations.Equilibria exhibit contrasting limits. One limit, at high cortical activation, generates widespread zero-lag synchrony among excitatory cells, with partial suppression of noise. Excitatory and inhibitory cells approach zero-lag local correlation, with 1/4 cycle lag-correlation at greater distances of separation. The high-activation limit defines a correlated system of attractor basins, capable of co-ordinating synaptic modifications and intracortical signal generation. Suppression of noise would enhance convergence about attractor basins in the manner of simulated annealing, while, conversely, the persistence of some noise prevents network paralysis by phase locking. At the opposite limit—that of low activation—spikes and waves have low cross- and auto-correlation, but have wide-spectrum sensitivity to inputs. It is hypothesised that cortical regions, transiently at equilibrium near these extremes, engage in interaction with each other and with subcortical systems, to generate ongoing sequences of attention and cognition.This account is compatible with classical and recently observed experimental phenomena. The principle features inferred from a simplified linear mathematical account are reproduced in a more physiologically realistic and non-linear numerical simulation.     

Erratum to: Attractor Dynamics and Thermodynamic Analogies in the Cerebral Cortex: Synchronous Oscillation,the Background EEG,and the Regulation of Attention

Ongoing changes in attention and cognition depend upon cortical/subcortical interactions, which select sequences of different spatial patterns of activation in the cortex.     

Gape Limitation, Prey Size Refuges and the Top–down Impacts of Piscivorous Largemouth Bass in Shallow Pond Ecosystems

Top–down control of phytoplankton biomass through piscivorous fish manipulation has been explored in numerous ecological and biomanipulation experiments. Piscivores are gape-limited predators and it is hypothesized that the distribution of gape sizes relative to distribution of body depths of prey fish may restrict piscivore effects cascading to plankton. We examined the top–down effects of piscivorous largemouth bass on nutrients, turbidity, phytoplankton, zooplankton and fish in ponds containing fish assemblages with species representing a range of body sizes and feeding habits (western mosquitofish, bluegill, channel catfish, gizzard shad and common carp). The experimental design consisted of three replicated treatments: fishless ponds (NF), fish community without largemouth bass (FC), and fish community with largemouth bass (FCB). Turbidity, chlorophyll a, cyclopoid copepodid and copepod nauplii densities were significantly greater in FC and FCB ponds than in NF ponds. However, these response variables were not significantly different in FC and FCB ponds. The biomass and density of shallow-bodied western mosquitofish were reduced and bluegill body depths shifted toward larger size classes in the presence of largemouth bass, but the biomass and density of all other fish species and of the total fish community were unaffected by the presence of largemouth bass. Our results show that top–down impacts of largemouth bass in ecosystems containing small- and deep-bodied fish species may be most intense at the top of the food web and alter the size distribution and species composition of the fish community. However, these top–down effects may not cascade to the level of the plankton when large-bodied benthivorous fish species are abundant.     

Mathematical Study of the Role of Gametocytes and an Imperfect Vaccine on Malaria Transmission Dynamics

A mathematical model is developed to assess the role of gametocytes (the infectious sexual stage of the malaria parasite) in malaria transmission dynamics in a community. The model is rigorously analysed to gain insights into its dynamical features. It is shown that, in the absence of disease-induced mortality, the model has a globally-asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold, known as the basic reproduction number (denoted by ℛ₀), is less than unity. Further, it has a unique endemic equilibrium if ℛ₀>1. The model is extended to incorporate an imperfect vaccine with some assumed therapeutic characteristics. Theoretical analyses of the model with vaccination show that an imperfect malaria vaccine could have negative or positive impact (in reducing disease burden) depending on whether or not a certain threshold (denoted by ∇) is less than unity. Numerical simulations of the vaccination model show that such an imperfect anti-malaria vaccine (with a modest efficacy and coverage rate) can lead to effective disease control if the reproduction threshold (denoted by ℛ_vac) of the disease is reasonably small. On the other hand, the disease cannot be effectively controlled using such a vaccine if ℛ_vac is high. Finally, it is shown that the average number of days spent in the class of infectious individuals with higher level of gametocyte is critically important to the malaria burden in the community.     

Structure of the 30 kDa HIV-1 RNA Dimerization Signal by a Hybrid Cryo-EM,NMR, and Molecular Dynamics Approach

1. Download high-res image (208KB)
2. Download full-size image

     

Adaptive Dynamics of Altruistic Cooperation in a Metapopulation: Evolutionary Emergence of Cooperators and Defectors or Evolutionary Suicide?

We investigate the evolution of public goods cooperation in a metapopulation model with small local populations, where altruistic cooperation can evolve due to assortment and kin selection, and the evolutionary emergence of cooperators and defectors via evolutionary branching is possible. Although evolutionary branching of cooperation has recently been demonstrated in the continuous snowdrift game and in another model of public goods cooperation, the required conditions on the cost and benefit functions are rather restrictive, e.g., altruistic cooperation cannot evolve in a defector population. We also observe selection for too low cooperation, such that the whole metapopulation goes extinct and evolutionary suicide occurs. We observed intuitive effects of various parameters on the numerical value of the monomorphic singular strategy. Their effect on the final coexisting cooperator–defector pair is more complex: changes expected to increase cooperation decrease the strategy value of the cooperator. However, at the same time the population size of the cooperator increases enough such that the average strategy does increase. We also extend the theory of structured metapopulation models by presenting a method to calculate the fitness gradient in a general class of metapopulation models, and try to make a connection with the kin selection approach.     

Effects of Temperature and Turbulence on the Predator–Prey Interactions between a Heterotrophic Flagellate and a Marine Bacterium

Biotic and abiotic factors can influence interactions between microbial grazers and their prey, thus impacting both the cycling of biogenic carbon within the surface layer of the ocean and the export of carbon to the deep ocean and higher trophic levels. In this study, microcosm experiments were used to evaluate the combined effect of temperature and turbulence on the growth rate of a marine bacterium (Vibrio splendidus), a protistan predator (Paraphysomonas sp.), and the community grazing impact of Paraphysomonas sp. on V. splendidus. It was found that the artificial turbulence generated (1.35 × 10⁻¹ cm² s⁻³) significantly increased the rates of growth of Paraphysomonas sp. at high (>10°C), but not low (<5°C) temperatures, and that turbulence had no effect on the growth of V. splendidus. Both flagellate and bacterial growth were temperature dependent and decreased 4- to 6-fold as temperatures decreased from 15 to 0°C. Bacterial grazing mortality by Paraphysomonas sp. was 1.3- to 2.5-fold greater in the turbulent than static treatments among all four temperatures, and the rates of cell-specific ingestion of bacteria by Paraphysomonas sp. was 2-fold greater at 15 and 10°C in the turbulent than in the static treatment. Hence, this study shows that turbulence can influence nanoflagellate grazing at temperatures >5°C and suggests that at low temperatures, increased viscosity may limit the size of organisms that can be affected by small-scale turbulence.Present address (M.P. Delaney): Goddard Earth Sciences & Technology Center, NASA Goddard Space Flight Center, Mail Code 900.1, Greenbelt, MD 20771     

Anthropogenic Landscape in Southeastern Amazonia: Contemporary Impacts of Low-Intensity Harvesting and Dispersal of Brazil Nuts by the Kayapó Indigenous People

Brazil nut, the Bertholletia excelsa seed, is one of the most important non-timber forest products in the Amazon Forest and the livelihoods of thousands of traditional Amazonian families depend on its commercialization. B. excelsa has been frequently cited as an indicator of anthropogenic forests and there is strong evidence that past human management has significantly contributed to its present distribution across the Amazon, suggesting that low levels of harvesting may play a positive role in B. excelsa recruitment. Here, we evaluate the effects of Brazil nut harvesting by the Kayapó Indigenous people of southeastern Amazonia on seedling recruitment in 20 B. excelsa groves subjected to different harvesting intensities, and investigated if management by harvesters influences patterns of B. excelsa distribution. The number of years of low-intensity Brazil nut harvesting by the Kayapó over the past two decades was positively related to B. excelsa seedling density in groves. One of the mechanisms behind the higher seedling density in harvested sites seems to be seed dispersal by harvesters along trails. The Kayapó also intentionally plant B. excelsa seeds and seedlings across their territories. Our results show not only that low-intensity Brazil nut harvesting by the Kayapó people does not reduce recruitment of seedlings, but that harvesting and/or associated activities conducted by traditional harvesters may benefit B. excelsa beyond grove borders. Our study supports the hypothesis that B. excelsa dispersal throughout the Amazon was, at least in part, influenced by indigenous groups, and strongly suggests that current human management contributes to the maintenance and formation of B. excelsa groves. We suggest that changes in Brazil nut management practices by traditional people to prevent harvesting impacts may be unnecessary and even counterproductive in many areas, and should be carefully evaluated before implementation.     

Interactions Between the Immune System and Cancer: A Brief Review of Non-spatial Mathematical Models

We briefly review spatially homogeneous mechanistic mathematical models describing the interactions between a malignant tumor and the immune system. We begin with the simplest (single equation) models for tumor growth and proceed to consider greater immunological detail (and correspondingly more equations) in steps. This approach allows us to clarify the necessity for expanding the complexity of models in order to capture the biological mechanisms we wish to understand. We conclude by discussing some unsolved problems in the mathematical modeling of cancer-immune system interactions.     

Modelling the Transmission Dynamics and Control of the Novel 2009 Swine Influenza (H1N1) Pandemic

The paper presents a deterministic compartmental model for the transmission dynamics of swine influenza (H1N1) pandemic in a population in the presence of an imperfect vaccine and use of drug therapy for confirmed cases. Rigorous analysis of the model, which stratifies the infected population in terms of their risk of developing severe illness, reveals that it exhibits a vaccine-induced backward bifurcation when the associated reproduction number is less than unity. The epidemiological consequence of this result is that the effective control of H1N1, when the reproduction number is less than unity, in the population would then be dependent on the initial sizes of the subpopulations of the model. For the case where the vaccine is perfect, it is shown that having the reproduction number less than unity is necessary and sufficient for effective control of H1N1 in the population (in such a case, the associated disease-free equilibrium is globally asymptotically stable). The model has a unique endemic equilibrium when the reproduction number exceeds unity. Numerical simulations of the model, using data relevant to the province of Manitoba, Canada, show that it reasonably mimics the observed H1N1 pandemic data for Manitoba during the first (Spring) wave of the pandemic. Further, it is shown that the timely implementation of a mass vaccination program together with the size of the Manitoban population that have preexisting infection-acquired immunity (from the first wave) are crucial to the magnitude of the expected burden of disease associated with the second wave of the H1N1 pandemic. With an estimated vaccine efficacy of approximately 80%, it is projected that at least 60% of Manitobans need to be vaccinated in order for the effective control or elimination of the H1N1 pandemic in the province to be feasible. Finally, it is shown that the burden of the second wave of H1N1 is expected to be at least three times that of the first wave, and that the second wave would last until the end of January or early February, 2010.     

Population Dynamics of the Critically Endangered Golden Lancehead Pitviper,Bothrops insularis: Stability or Decline?

Little is known about vital rates of snakes generally because of the difficulty in collecting data. Here we used a robust design mark-recapture model to estimate survival, behavioral effects on capture probability, temporary emigration, abundance and test the hypothesis of population decline in the golden lancehead pitviper, Bothrops insularis, an endemic and critically endangered species from southeastern Brazil. We collected data at irregular intervals over ten occasions from 2002 to 2010. Survival was slightly higher in the wet season than in the dry season. Temporal emigration was high, indicating the importance of accounting for this parameter both in the sampling design and modeling. No behavioral effects were detected on capture probability. We detected an average annual population decrease ( = 0.93, CI = 0.47–1.38) during the study period, but estimates included high uncertainty, and caution in interpretation is needed. We discuss the potential effects of the illegal removal of individuals and the implications of the vital rates obtained for the future persistence and conservation of this endemic, endangered species.