Global dynamics of a novel delayed logistic equation arising from cell biology

The delayed logistic equation (also known as Hutchinson's equation or Wright's equation) was originally introduced to explain oscillatory phenomena in ecological dynamics. While it motivated the development of a large number of mathematical tools in the study of nonlinear delay differential equations, it also received criticism from modellers because of the lack of a mechanistic biological derivation and interpretation. Here we propose a new delayed logistic equation, which has clear biological underpinning coming from cell population modelling. This nonlinear differential equation includes terms with discrete and distributed delays. The global dynamics is completely described, and it is proven that all feasible nontrivial solutions converge to the positive equilibrium. The main tools of the proof rely on persistence theory, comparison principles and an $L^2$-perturbation technique. Using local invariant manifolds, a unique heteroclinic orbit is constructed that connects the unstable zero and the stable positive equilibrium, and we show that these three complete orbits constitute the global attractor of the system. Despite global attractivity, the dynamics is not trivial as we can observe long-lasting transient oscillatory patterns of various shapes. We also discuss the biological implications of these findings and their relations to other logistic type models of growth with delays

Modelling Walleye Population and Its Cannibalism Effect

Walleye is a very common recreational fish in Canada with a strong cannibalism tendency, such that walleyes with larger sizes will consume their smaller counterparts when food sources are limited or a surplus of adults is present. Cannibalism may be a factor promoting population oscillation. As fish reach a certain age or biological stage (i.e. biological maturity), the number of fish achieving that stage is known as fish recruitment. The objective of this thesis is to model the walleye population with its recruitment and cannibalism effect. A matrix population model has been introduced to characterize the walleye population into three different groups: newborns, juveniles, and adults. A delay differential equation (DDE) model has also been introduced to characterize walleyes into two groups including juveniles and adults. Local and global stabilities of equilibria have been discussed in both models. Furthermore, numerical simulations are present to visualize the effects of both models

Global asymptotic stability for neural network models with distributed delays

In this paper, we obtain the global asymptotic stability of the zero solution of a general n-dimensional delayed differential system, by imposing a condition of dominance of the nondelayed terms which cancels the delayed effect. We consider several delayed differential systems in general settings, which allow us to study, as subclasses, the well known neural network models of Hopfield, Cohn-Grossberg, bidirectional associative memory, and static with S-type distributed delays. For these systems, we establish sufficient conditions for the existence of a unique equilibrium and its global asymptotic stability, without using the Lyapunov functional technique. Our results improve and generalize some existing ones.Fundação para a Ciência e a Tecnologia (FCT

Global asymptotic stability of a scalar delay Nicholson's blowflies equation in periodic environment

This paper is considered with a scalar delay Nicholson’s blowflies equation in periodic environment. By taking advantage of some novel differential inequality techniques and the fluctuation lemma, we set up the sharp condition to characterize the global asymptotic stability of positive periodic solutions on the addressed equation. The obtained results improve and supplement some existing ones in recent literature, and then give a more perfect answer to an open problem proposed by Berezansky et al. in [Appl. Math. Model. 34(2010), 1405–1417]. In particular, two numerical examples are provided to verify the reliability and feasibility of the theoretical findings

Local and global stability for Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks

This paper addresses the local and global stability of n-dimensional Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks. Necessary and sufficient conditions for local stability independent of the choice of the delay functions are given, by imposing a weak nondelayed diagonal dominance which cancels the delayed competition effect. The global asymptotic stability of positive equilibria is established under conditions slightly stronger than the ones required for the linear stability. For the case of monotone interactions, however, sharper conditions are presented. This paper generalizes known results for discrete delays to systems with distributed delays. Several applications illustrate the results.Fundação para a Ciência e a Tecnologia (FCT) - programa POCI, projecto PDCT/ MAT/56476/2004.Portugal-FEDE

Studying Both Direct and Indirect Effects in Predator-Prey Interaction

Studying and modelling the interaction between predators and prey have been one of the central topics in ecology and evolutionary biology. In this thesis, we study two different aspects of predator-prey interaction: direct effect and indirect effect. Firstly, we study the direct predation between predators and prey in a patchy landscape. Secondly, we study indirect effects between predators and prey. Thirdly, we extend our previous model by incorporating a stage-structure into prey. Finally, we further extend our previous model by incorporating spatial structures into modelling

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)