Mathematical Analysis for a Discrete Predator-Prey Model with Time Delay and Holling II Functional Response

This paper is concerned with a discrete predator-prey model with Holling II functional response and delays. Applying Gaines and Mawhin's continuation theorem of coincidence degree theory and the method of Lyapunov function, we obtain some sufficient conditions for the existence global asymptotic stability of positive periodic solutions of the model

Dynamic behaviors of a delay differential equation model of plankton allelopathy

AbstractIn this paper, we consider a modified delay differential equation model of the growth of n-species of plankton having competitive and allelopathic effects on each other. We first obtain the sufficient conditions which guarantee the permanence of the system. As a corollary, for periodic case, we obtain a set of delay-dependent condition which ensures the existence of at least one positive periodic solution of the system. After that, by means of a suitable Lyapunov functional, sufficient conditions are derived for the global attractivity of the system. For the two-dimensional case, under some suitable assumptions, we prove that one of the components will be driven to extinction while the other will stabilize at a certain solution of a logistic equation. Examples show the feasibility of the main results

Using Delay-Differential Equations for Modeling Calcium Cycling in Cardiac Myocytes

The cycling of calcium at the intracellular level of cardiac cells plays a key role in the excitation-contraction process. The interplay between ionic currents, buffering agents, and calcium release from the sarcoplasmic reticulum (SR) is a complex system that has been shown experimentally to exhibit complex dynamics including period-2 states (alternans) and higher-order rhythms. Many of the calcium cycling activities involve the sensing, binding, or diffusion of calcium between intracellular compartments; these are physical processes that take time and typically are modeled by “relaxation” equations where the steady-state value and time course of a particular variable are specified through an ordinary differential equation (ODE) with a time constant. An alternative approach is to use delay-differential equations (DDEs), where the delays in the system correspond to non-instantaneous events. In this thesis, we present a thorough overview of results from calcium cycling experiments and proposed intracellular calcium cycling models, as well as the context of alternans and delay-differential equations in cardiac modeling. We utilize a DDE to model the diffusion of calcium through the SR by replacing the relaxation ODE typically used for this process. The relaxation time constant τa is replaced by a delay δj, which could also be interpreted as the refractoriness of ryanodine receptor channels after releasing calcium from the sarcoplasmic reticulum. This is the first application of delay-differential equations to modeling calcium cycling dynamics, and to modeling cardiac systems at the cellular level. We analyzed the dynamical behaviors of the system and focus on the factors that have been shown to produce alternans and irregular dynamics in experiments and models with cardiac myocytes. We found that chaotic calcium dynamics could occur even for a more physiologically revelant SR calcium release slope than comparable ODE models. Increasing the SR release slope did not affect the calcium dynamics, but only shifted behavior down to lower values of the delay, allowing alternans, higher-order behavior, and chaos to occur for smaller delays than in simulations with a normal SR release slope. For moderate values of the delay, solely alternans and 1:1 steady-state behavior were observed. Above a particular threshold value for the delay, chaos appeared in the dynamics and further increasing the delay caused the system to destabilize under broader ranges of periods. We also compare our results with other models of intracellular calcium cycling and suggest promising avenues for further development of our preliminary work

The paradox of the plankton: Investigating the effect of inter-species competition of phytoplankton and its sensitivity to nutrient supply and external forcing

Hutchinson (1961) first posed the paradox of the plankton: Why do so many phytoplankton species coexist while competing for a limited number of resources? High biodiversity has been explained in terms of the phytoplankton system not reaching an equilibrium state. Spatial and temporal variability can be achieved through externally imposed physical variability or internally-induced behaviour including periodic oscillations or irregular, chaotic behaviour. The research presented in this thesis investigates whether the non-equilibrium, chaotic response of the phytoplankton community is a likely outcome within the aquatic ecosystems. The thesis addresses the extent that chaotic behaviour remains a robust response with externally-imposed environmental variability. The sparsity of long-term time-series data and infrequent sampling inhibits the ability to verify whether marine ecosystems exhibit complex behaviour. The analysis of the time-series records of phytoplankton taxa in the English Channel suggests that chaos might occur within diatom and dinoflagellates abundance time series. However, simulations using a chemostat model for phytoplankton and nutrients suggests that time series sampled every 1-2 days for more than 5 years are required to confidently distinguish deterministic chaos from noise. The model simulations suggest that the community response depends on the phytoplankton requirement for nutrients and attributed physiological traits allowing each species to be a stronger competitor for a different resource. A wider inter-species specialization increases the likelihood of oscillatory and chaotic responses, with competitive exclusion decreasing from 50% to 20% of the cases. Higher departures from the Redfield ratio in the elemental composition of species favour complex community behaviour and act to increase biodiversity. Whether chaotic response can be sustained is sensitive to the strength of the diffusive feedback between nutrient supply and ambient nutrient concentration that acts to sustain steady-state nutrient concentrations. Including seasonal and stochastic variability in the nutrient supply reveals that the frequency of chaotic dynamics increases by 20% and 45% respectively. In addition, seasonal forcing leads to temporal variability in the strength of the chaotic response, with chaos becoming more prevalent in the summer. In contrast to a well-mixed, homogeneous environment, physical dispersal can stir different phytoplankton communities together, which might act to inhibit chaos, but at the same time enhance phytoplankton diversity. Idealised model simulations are conducted to mimic the small and large scale transport processes by including 2 or 3 well-mixed boxes. Locally generated chaotic response is sustained if: 1) there is a low rate of exchange with a strong nutrient competitor that maintains the contrasts in the community structure; 2) a strong competitor is inhibited by a high mortality rate. In addition, if the local community is outcompeted, chaos can be exported through the advection of stronger competitors that exhibit chaotic fluctuations. This study highlights the importance of understanding the interactions between ambient nutrients and phytoplankton community. The variability in the nutrient supply and connectivity between ecosystems shape the community response to inter-species competition. Complex behaviour arising from inter-species competition is suggested to have a significant contribution in driving biodiversity. Future research on assessing the extent of chaos requires extending and analysing the available time-series data obtained from stable or isolated marine provinces

Asymptotic stability for population models and neural networks with delays

Tese de doutoramento em Matemática (Análise Matemática), apresentada à Universidade de Lisboa através da Faculdade de Ciências, 2008In this thesis, the global asymptotic stability of solutions of several functional differential equations is addressed, with particular emphasis on the study of global stability of equilibrium points of population dynamics and neural network models. First, for scalar retarded functional differential equations, we use weaker versions of the usual Yorke and 3/2-type conditions, to prove the global attractivity of the trivial solution. Afterwards, we establish new sufficient conditions for the global attractivity of the positive equilibrium of a general scalar delayed population model, and illustrate the situation applying these results to two food-limited population models with delays. Second, for n-dimensional Lotka-Volterra systems with distributed delays, the local and global stability of a positive equilibrium, independently of the choice of the delay functions, is addressed assuming that instantaneous negative feedbacks are present. Finally, we obtain the existence and global asymptotic stability of an equilibrium point of a general neural network model by imposing a condition of dominance of the nondelayed terms. The generality of the model allows us to study, as particular situations, the neural network models of Hop_eld, Cohn-Grossberg, bidirectional associative memory, and static with S-type distributed delays. In our proofs, we do not use Lyapunov functionals and our method applies to general delayed di_erential equations.Nesta tese estuda-se a estabilidade global assimptótica de soluções de equações diferenciais funcionais que, pela generalidade com que são apresentadas, possuem uma vasta aplicabilidade em modelos de dinâmica de populações e em modelos de redes neuronais. Numa primeira fase, para equações diferenciais funcionais escalares retardadas, assumem-se novas versões das condições de Yorke e tipo 3/2 para provar a atractividade global da solução nula. Seguidamente, aplicam-se os resultados obtidos a um modelo geral de dinâmica de populações escalar com atrasos, obtendo-se condições suficientes para a atractividade global de um ponto de equilíbrio positivo, e ilustra-se a situação com o estudo de dois modelos conhecidos. Numa segunda fase, para sistemas n-dimensionais de tipo Lotka-Volterra com atrasos distribuídos, estuda-se a estabilidade local e global de um ponto de equilíbrio positivo (caso exista) assumindo condições de dominância dos termos com atrasos pelos termos sem atrasos. Por último, novamente assumindo condições de donimância, obtém-se a existência e estabilidade global assimptótica de um ponto de equilíbrio para um modelo geral de redes neuronais com atrasos. A generalidade do modelo estudado permite obter, como situações particulares, critérios de estabilidade global para modelos de redes neuronais de Hopfield, de Cohn-Grossberg, modelos de memória associativa bidireccional e modelos estáticos com atrasos distribuídos tipo-S. De referir que as demonstrações apresentadas não envolvem o uso de funcionais de Lyapunov, o que permite obter critérios de estabilidade para equações diferenciais funcionais bastante gerais.Universidade do Minho (UM), Departamento de Matemática (DMAT), Centro de Matemática (CMAT); Fundação para a Ciência e Tecnologia (FCT)

Collective Information Processing and Criticality, Evolution and Limited Attention.

Im ersten Teil analysiere ich die Selbstorganisation zur Kritikalität (hier ein Phasenübergang von Ordnung zu Unordnung) und untersuche, ob Evolution ein möglicher Organisationsmechanismus ist. Die Kernfrage ist, ob sich ein simulierter kohäsiver Schwarm, der versucht, einem Raubtier auszuweichen, durch Evolution selbst zum kritischen Punkt entwickelt, um das Ausweichen zu optimieren? Es stellt sich heraus, dass (i) die Gruppe den Jäger am besten am kritischen Punkt vermeidet, aber (ii) nicht durch einer verstärkten Reaktion, sondern durch strukturelle Veränderungen, (iii) das Gruppenoptimum ist evolutionär unstabiler aufgrund einer maximalen räumlichen Selbstsortierung der Individuen. Im zweiten Teil modelliere ich experimentell beobachtete Unterschiede im kollektiven Verhalten von Fischgruppen, die über mehrere Generationen verschiedenen Arten von größenabhängiger Selektion ausgesetzt waren. Diese Größenselektion soll Freizeitfischerei (kleine Fische werden freigelassen, große werden konsumiert) und die kommerzielle Fischerei mit großen Netzbreiten (kleine/junge Individuen können entkommen) nachahmen. Die zeigt sich, dass das Fangen großer Fische den Zusammenhalt und die Risikobereitschaft der Individuen reduziert. Beide Befunde lassen sich mechanistisch durch einen Aufmerksamkeits-Kompromiss zwischen Sozial- und Umweltinformationen erklären. Im letzten Teil der Arbeit quantifiziere ich die kollektive Informationsverarbeitung im Feld. Das Studiensystem ist eine an sulfidische Wasserbedingungen angepasste Fischart mit einem kollektiven Fluchtverhalten vor Vögeln (wiederholte kollektive Fluchttauchgängen). Die Fische sind etwa 2 Zentimeter groß, aber die kollektive Welle breitet sich über Meter in dichten Schwärmen an der Oberfläche aus. Es zeigt sich, dass die Wellengeschwindigkeit schwach mit der Polarisation zunimmt, bei einer optimalen Dichte am schnellsten ist und von ihrer Richtung relativ zur Schwarmorientierung abhängt.In the first part, I focus on the self-organization to criticality (here an order-disorder phase transition) and investigate if evolution is a possible self-tuning mechanism. Does a simulated cohesive swarm that tries to avoid a pursuing predator self-tunes itself by evolution to the critical point to optimize avoidance? It turns out that (i) the best group avoidance is at criticality but (ii) not due to an enhanced response but because of structural changes (fundamentally linked to criticality), (iii) the group optimum is not an evolutionary stable state, in fact (iv) it is an evolutionary accelerator due to a maximal spatial self-sorting of individuals causing spatial selection. In the second part, I model experimentally observed differences in collective behavior of fish groups subject to multiple generation of different types of size-dependent selection. The real world analog to this experimental evolution is recreational fishery (small fish are released, large are consumed) and commercial fishing with large net widths (small/young individuals can escape). The results suggest that large harvesting reduces cohesion and risk taking of individuals. I show that both findings can be mechanistically explained based on an attention trade-off between social and environmental information. Furthermore, I numerically analyze how differently size-harvested groups perform in a natural predator and fishing scenario. In the last part of the thesis, I quantify the collective information processing in the field. The study system is a fish species adapted to sulfidic water conditions with a collective escape behavior from aerial predators which manifests in repeated collective escape dives. These fish measure about 2 centimeters, but the collective wave spreads across meters in dense shoals at the surface. I find that wave speed increases weakly with polarization, is fastest at an optimal density and depends on its direction relative to shoal orientation

Dynamical models in neuroscience: the delay FitzHugh-Nagumo equation

Il primo modello matematico in grado di descrivere il prototipo di un sistema eccitabile assimilabile ad un neurone fu sviluppato da R. FitzHugh e J. Nagumo nel 1961. Tale modello, per quanto schematico, rappresenta un importante punto di partenza per la ricerca nell'ambito neuroscientifico delle dinamiche neuronali, ed è infatti capostipite di una serie di lavori che hanno puntato a migliorare l’accuratezza e la predicibilità dei modelli matematici per le scienze. L’elevato grado di complessità nello studio dei neuroni e delle dinamiche inter-neuronali comporta, tuttavia, che molte delle caratteristiche e delle potenzialità dell’ambito non siano ancora state comprese appieno. In questo lavoro verrà approfondito un modello ispirato al lavoro originale di FitzHugh e Nagumo. Tale modello presenta l’introduzione di un termine di self-coupling con ritardo temporale nel sistema di equazioni differenziali, diventa dunque rappresentativo di modelli di campo medio in grado di descrivere gli stati macroscopici di un ensemble di neuroni. L'introduzione del ritardo è funzionale ad una descrizione più realistica dei sistemi neuronali, e produce una dinamica più ricca e complessa rispetto a quella presente nella versione originale del modello. Sarà mostrata l'esistenza di una soluzione a ciclo limite nel modello che comprende il termine di ritardo temporale, ove tale soluzione non può essere interpretata nell’ambito delle biforcazioni di Hopf. Allo scopo di esplorare alcune delle caratteristiche basilari della modellizzazione del neurone, verrà principalmente utilizzata l’impostazione della teoria dei sistemi dinamici, integrando dove necessario con alcune nozioni provenienti dall’ambito fisiologico. In conclusione sarà riportata una sezione di approfondimento sulla integrazione numerica delle equazioni differenziali con ritardo

Mental and sensorimotor extrapolation fare better than motion extrapolation in the offset condition

Evidence for motion extrapolation at motion offset is scarce. In contrast, there is abundant evidence that subjects mentally extrapolate the future trajectory of weak motion signals at motion offset. Further, pointing movements overshoot at motion offset. We believe that mental and sensorimotor extrapolation is sufficient to solve the problem of perceptual latencies. Both present the advantage of being much more flexible than motion extrapolatio