Logistic Population Growth and Beyond: The Influence of Advection and Nonlocal Effects

Abstract

By introducing the logistic equation in the context of demographic modelling [1], J.F. Verhulst made seminal contributions to at least two important fields of research: On the one hand, to the quantitative approach to Population Dynamics, a subject in which substantial development beyond Verhulst ideas only occurred 100 years after him. On the other, to the basics of Nonlinear Science through a detailed study of the predictions of a nonlinear model and its comparison with experimental data. The dynamics of biological populations in aquatic environments [2, 3, 4] is an excellent framework to see recent developments in which these disciplines work together. Growth, limitation, competition, predation and all the other kinds of biological interactions appear in this context, and nonlinear processes are relevant both in the biological dynamics and in the motion of the turbulent fluid in which the population lives. In this contribution we present two examples in the above field. In both cases a prominent role is played by the logistic growth process (i.e. population growth limited by finite resources), but other ingredients are also included that strongly change the phenomenology. In Section 2, a phytoplankton population experiencing logistic growth is studied, but in interaction with zooplankton predators that maintain it in a state below the carrying capacity of the supporting medium. In the appropriate parameter regime the system behaves in an excitable way, with perturbations inducing large excitationdeexcitation cycles of the phytoplankton population. The excitation cycles become strongly affected by the presence of chaotic motion of the fluid containing the populations. In section 3, an individual based model of interacting organisms is presented, for which logistic growth is again the main ingredient. Reproduction of a given individual is limited by the presence of others in a neighborhood of finite size. This nonlocal character of the interaction is enough to produce an instability of the basic state of particles homogenously distributed, and clustering of the individuals occurs, which form groups arranged in an hexagonal lattice (when the population lives in a two-dimensional space).