Mathematical Models in Population Dynamics and Ecology

Abstract

We introduce the most common quantitative approaches to population dynamics and ecology, emphasizing the different theoretical foundations and assumptions. These populations can be aggregates of cells, simple unicellular organisms, plants or animals.The basic types of biological interactions are analysed: consumer-resource, prey-predation, competition and mutualism. Some of the modern developments associated with the concepts of chaos, quasi-periodicity, and structural stability are discussed. To describe short-and long-range population dispersal, the integral equation approach is derived, and some of its consequences are analysed. We derive the standard McKendrick age-structured density dependent model, and a particular solution of the McKendrick equation is obtained by elementary methods. The existence of demography growth cycles is discussed, and the differences between mitotic and sexual reproduction types are analysed.

On the Weak Solutions of the McKendrick Equation: Existence of Demography Cycles

“…To calculate explicitly the general solutions of the partial differential equation (2.1) obeying the boundary condition (2.4) and prescribed initial data, we use the technique of characteristics, [12] and [5]. Writing equation (2.1) in the form, dn(a,t) dt = −µ(a)n(a, t), the solutions of (2.1) are also solutions of the system of ordinary differential equations,…”

Equilibrium price dynamics in an overlapping-generations exchange economy

“…The existence of solutions of the Cauchy problem for the linear equation (2) together with the boundary condition (3) is well established by semigroup techniques and by the method of characteristics. For reviews see, for example, Webb (1985), Cushing (1998) and Dilão (2006). The population density at time t is determined from the initial population density, n(a, t = 0) = ψ(a), with a, t ∈ R + .…”

“…The population density at time t is determined from the initial population density, n(a, t = 0) = ψ(a), with a, t ∈ R + . According to the standard theory of first order partial differential equations, the characteristic curves of the McKendrick equation are the solutions of the differential equation da dt = 1, being straight lines with equation, a − a 0 = t − t 0 , Dilão (2006). Therefore, as dn dt = −µ(a)n, within a characteristic curve, the solutions of the McKendrick equation can be written as,…”

Population dynamics, life stage and ecological modeling in Chrysomya albiceps (Wiedemann) (Diptera: Calliphoridae)

“…Within each age class, the individuals of a species behave differently, and have different types of dependence on the environment and resource needs (Cushing et al 2003, Dilão 2006. Several studies have attempted to investigate age-structure problems by examining life tables in which the age of individual insects is determined and used to assess the standing population age structure (Gabre et al 2005, Liu et al 2008.…”

“…To describe a population with age classes or life-stages, a discrete formalism can be adopted, where the transition between different age classes or stages is described in a matrix form. One of the advantages of this type of model is that it can be naturally related to ield or laboratory data (Caswell 2001, Dilão 2006. The Leslie matrix model (Leslie 1945(Leslie , 1948 has been frequently used to describe the dynamics of age-or stagestructured populations.…”