Logistic Growth Term Research Articles

Optimal control problems of nonlinear degenerate parabolic differential systems with logistic time-varying delays

In this paper, we study a bioeconomic model for optimal control problems for a class of systems governed by degenerate parabolic equations governing diffusive biological species with logistic growth terms and multiple time-varying delays. We prove the existence, uniqueness and regularity results for this degenerate parabolic equation. The viscosity solution theory is used to obtain the existence result. Afterwards, we formulate the optimal control problem in two cases. Firstly, we suppose that this biological species causes damage to environment (forest lands, farming, etc): the optimal control is the trapping rate and the cost functional is a combination of damage and trapping costs. Secondly, we consider the optimal harvesting control of a biological species: the optimal control is a distribution of harvesting effort on the biological species and the cost functional measure the difference between economic revenue and cost. The existence and the condition of uniqueness of the optimal solution are derived. First-order necessary conditions of optimality are obtained.

Addendum to "Random population dispersal in a linear hostile environment".

We extend the previous results, describing the population dispersal that occurs in some insects and small animal populations when this process is not strictly random, by including both the downgradient diffusion and the full Pearl-Verhulst logistic growth term in the equation of evolution. Motivated by the increasing fragmentation of natural habitats that is the result of human activities, we consider a finite habitat surrounded by a hostile environment. Previous work [Phys Rev. E 62, 4032 (2000)] considered only the case of an unbounded habitat, obviating issues concerned with the critical habitat size and the adoption of strategies best suited to achieve lower densities by dispersal through downgradient diffusion.

A Simplified Model of Spatiotemporal Population Dynamics

This paper is an extension of the model of population growth and migration originally developed by H. Hotelling in 1921. This model consists of two ingredients, a logistic growth function and a linear spatial diffusion term. The author notes that the saturation population can be affected by the development of new technology and that improvements in transportation have increased the possibilities for migration. "Basic nonlinearities are introduced by use of a production technology with increasing-decreasing returns to scale. It is demonstrated how industrial takeoffs, population transitions, and agglomerative spatial patterns can emerge by changing the model parameters."

Diffusive logistic growth in deterministic and stochastic environments

A study is made of a non-linear diffusion equation with logistic growth terms. The equation is solvable with space and time dependent carrying capacities. Examples of varying complexity are presented, some of which involve deterministic carrying capacities, others involve carrying capacities whose parameters are random functions of time or space. Fluctuations in environmental parameters are characterized by non-white correlated noise. Solutions are given in terms of correlation times or correlation lengths. Explicit solutions are developed for the deterministic cases, while the stochastic cases are solved by using the beta distribution as an approximation to the probability distribution function of the population density.