Age-dependent diffusive Lotka-Volterra type systems

In this paper it is shown that the sub-supersolution method works for age-dependent diffusive nonlinear systems with non-local initial conditions. As application, we prove the existence and uniqueness of positive solution for a kind of Lotka-Volterra systems, as well as the blow-up in finite time in a particular case.Ministerio de Ciencia y Tecnologí

Robust permanence for interacting structured populations

The dynamics of interacting structured populations can be modeled by $\frac{dx_i}{dt}= A_i (x)x_i$ where $x_i\in \R^{n_i}$, $x=(x_1,\dots,x_k)$, and $A_i(x)$ are matrices with non-negative off-diagonal entries. These models are permanent if there exists a positive global attractor and are robustly permanent if they remain permanent following perturbations of $A_i(x)$. Necessary and sufficient conditions for robust permanence are derived using dominant Lyapunov exponents $\lambda_i(\mu)$ of the $A_i(x)$ with respect to invariant measures $\mu$. The necessary condition requires $\max_i \lambda_i(\mu)>0$ for all ergodic measures with support in the boundary of the non-negative cone. The sufficient condition requires that the boundary admits a Morse decomposition such that $\max_i \lambda_i(\mu)>0$ for all invariant measures $\mu$ supported by a component of the Morse decomposition. When the Morse components are Axiom A, uniquely ergodic, or support all but one population, the necessary and sufficient conditions are equivalent. Applications to spatial ecology, epidemiology, and gene networks are given

Selected topics on reaction-diffusion-advection models from spatial ecology

We discuss the effects of movement and spatial heterogeneity on population dynamics via reaction-diffusion-advection models, focusing on the persistence, competition, and evolution of organisms in spatially heterogeneous environments. Topics include Lokta-Volterra competition models, river models, evolution of biased movement, phytoplankton growth, and spatial spread of epidemic disease. Open problems and conjectures are presented

Asymptotic behavior of age-structured and delayed Lotka-Volterra models

In this work we investigate some asymptotic properties of an age-structured Lotka-Volterra model, where a specific choice of the functional parameters allows us to formulate it as a delayed problem, for which we prove the existence of a unique coexistence equilibrium and characterize the existence of a periodic solution. We also exhibit a Lyapunov functional that enables us to reduce the attractive set to either the nontrivial equilibrium or to a periodic solution. We then prove the asymptotic stability of the nontrivial equilibrium where, depending on the existence of the periodic trajectory, we make explicit the basin of attraction of the equilibrium. Finally, we prove that these results can be extended to the initial PDE problem.Comment: 29 page

Dynamics of a structured slug population model in the absence of seasonal variation

We develop a novel, nonlinear structured population model for the slug Deroceras reticulatum, a highly significant agricultural pest of great economic impact, in both organic and non-organic settings. In the absence of seasonal variations, we numerically explore the effect of life history traits that are dependent on an individual's size and measures of population biomass. We conduct a systematic exploration of parameter space and highlight the main mechanisms and implications of model design. A major conclusion of this work is that strong size dependent predation significantly adjusts the competitive balance, leading to non-monotonic steady state solutions and slowly decaying transients consisting of distinct generational cycles. Furthermore, we demonstrate how a simple ratio of adult to juvenile biomass can act as a useful diagnostic to distinguish between predated and non-predated environments, and may be useful in agricultural settings

STOCHASTIC DELAY DIFFERENTIAL EQUATIONS WITH APPLICATIONS IN ECOLOGY AND EPIDEMICS

Mathematical modeling with delay differential equations (DDEs) is widely used for analysis and predictions in various areas of life sciences, such as population dynamics, epidemiology, immunology, physiology, and neural networks. The memory or time-delays, in these models, are related to the duration of certain hidden processes like the stages of the life cycle, the time between infection of a cell and the production of new viruses, the duration of the infectious period, the immune period, and so on. In ordinary differential equations (ODEs), the unknown state and its derivatives are evaluated at the same time instant. In DDEs, however, the evolution of the system at a certain time instant depends on the past history/memory. Introduction of such time-delays in a differential model significantly improves the dynamics of the model and enriches the complexity of the system. Moreover, natural phenomena counter an environmental noise and usually do not follow deterministic laws strictly but oscillate randomly about some average values, so that the population density never attains a fixed value with the advancement of time. Accordingly, stochastic delay differential equations (SDDEs) models play a prominent role in many application areas including biology, epidemiology and population dynamics, mostly because they can offer a more sophisticated insight through physical phenomena than their deterministic counterparts do. The SDDEs can be regarded as a generalization of stochastic differential equations (SDEs) and DDEs.This dissertation, consists of eight Chapters, is concerned with qualitative and quantitative features of deterministic and stochastic delay differential equations with applications in ecology and epidemics. The local and global stabilities of the steady states and Hopf bifurcations with respect of interesting parameters of such models are investigated. The impact of incorporating time-delays and random noise in such class of differential equations for different types of predator-prey systems and infectious diseases is studied. Numerical simulations, using suitable and reliable numerical schemes, are provided to show the effectiveness of the obtained theoretical results.Chapter 1 provides a brief overview about the topic and shows significance of the study. Chapter 2, is devoted to investigate the qualitative behaviours (through local and global stability of the steady states) of DDEs with predator-prey systems in case of hunting cooperation on predators. Chapter 3 deals with the dynamics of DDEs, of multiple time-delays, of two-prey one-predator system, where the growth of both preys populations subject to Allee effects, with a direct competition between the two-prey species having a common predator. A Lyapunov functional is deducted to investigate the global stability of positive interior equilibrium. Chapter 4, studies the dynamics of stochastic DDEs for predator-prey system with hunting cooperation in predators. Existence and uniqueness of global positive solution and stochastically ultimate boundedness are investigated. Some sufficient conditions for persistence and extinction, using Lyapunov functional, are obtained. Chapter 5 is devoted to investigate Stochastic DDEs of three-species predator prey system with cooperation among prey species. Sufficient conditions of existence and uniqueness of an ergodic stationary distribution of the positive solution to the model are established, by constructing a suitable Lyapunov functional. Chapter 6 deals with stochastic epidemic SIRC model with time-delay for spread of COVID-19 among population. The basic reproduction number ℛs0 for the stochastic model which is smaller than ℛ0 of the corresponding deterministic model is deduced. Sufficient conditions that guarantee the existence of a unique ergodic stationary distribution, using the stochastic Lyapunov functional, and conditions for the extinction of the disease are obtained. In Chapter 7, some numerical schemes for SDDEs are discussed. Convergence and consistency of such schemes are investigated. Chapter 8 summaries the main finding and future directions of research. The main findings, theoretically and numerically, show that time-delays and random noise have a significant impact in the dynamics of ecological and biological systems. They also have an important role in ecological balance and environmental stability of living organisms. A small scale of white noise can promote the survival of population; While large noises can lead to extinction of the population, this would not happen in the deterministic systems without noises. Also, white noise plays an important part in controlling the spread of the disease; When the white noise is relatively large, the infectious diseases will become extinct; Re-infection and periodic outbreaks can also occur due to the time-delay in the transmission terms