A simplified model for age-dependent population dynamics (original) (raw)
Discrete age-structured population model with age dependent harvesting and its stability analysis
Applied Mathematics and Computation, 2008
A discrete age-structured Bernardelli, Lewis and Leslie (BLL) population model for a single species is formulated under age dependent harvesting condition. Here, after an initial period during which the growth of the species is not much, harvesting is performed. Its rate is proportional to the available bio-mass (number of species) of different age group population and decreases with the age of the species. The modified Leslie matrix for the present models is derived. Stability of the system is studied from the ratio of the population densities at different times. As a particular case, not considering the harvesting of species, the classical age-structured population model of [J.N. Kapur, Stability analysis of continuous and discrete population models, Indian J. Pure Appl. Math. 9 (1976) 702-708] is obtained.
Optimization of harvesting age in an integral age-dependent model of population dynamics
Mathematical Biosciences, 2005
The paper deals with optimal control in a linear integral age-dependent model of population dynamics. A problem for maximizing the harvesting return on a finite time horizon is formulated and analyzed. The optimal controls are the harvesting age and the rate of population removal by harvesting. The gradient and necessary condition for an extremum are derived. A qualitative analysis of
Harvesting in a resource dependent age structured Leslie type population model
Mathematical Biosciences, 2004
We analyse the effect of harvesting in a resource dependent age structured population model, deriving the conditions for the existence of a stable steady state as a function of fertility coefficients, harvesting mortality and carrying capacity of the resources. Under the effect of proportional harvest, we give a sufficient condition for a population to extinguish, and we show that the magnitude of proportional harvest depends on the resources available to the population. We show that the harvesting yield can be periodic, quasi-periodic or chaotic, depending on the dynamics of the harvested population. For populations with large fertility numbers, small harvesting mortality leads to abrupt extinction, but larger harvesting mortality leads to controlled population numbers by avoiding over consumption of resources. Harvesting can be a strategy in order to stabilise periodic or quasi-periodic oscillations in the number of individuals of a population.
Mathematical Biosciences, 2007
The paper analyzes optimal harvesting of age-structured populations described by the Lotka–McKendrik model. It is shown that the optimal time- and age-dependent harvesting control involves only one age at natural conditions. This result leads to a new optimization problem with the time-dependent harvesting age as an unknown control. The integral Lotka model is employed to explicitly describe the time-varying age
Optimal harvesting for periodic age-dependent population dynamics with logistic term
Applied Mathematics and Computation, 2009
We prove an asymptotic behavior result for an age-dependent population dynamics with logistic term and periodic vital rates. We investigate next an optimal harvesting problem related to a periodic age-structured model with logistic term. Existence of an optimal control and necessary optimality conditions are established. A conceptual algorithm to approximate the optimal pair is derived and some numerical experiments are presented.
Mathematical Models in Population Dynamics and Ecology
Biomathematics, 2006
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.
Optimal Harvesting for a Nonlinear Age-Dependent Population Dynamics
Journal of Mathematical Analysis and Applications, 1998
Here we investigate an optimal harvesting problem for a nonlinear age-dependent population dynamics. Existence and uniqueness of a positive solution for the model are demonstrated. The structure of the solution is also investigated. We establish the existence of the optimal control and the convergence of a certain fractional step scheme. For some approximating problems we obtain the optimal controllers in feedback form via the dynamic programming method.