Population dynamics logistic equation

This is a difference equation widely used as a model in ecology and population dynamics (May (1974, 1987), Gleick (1987), Devaney (1992), Ott (1993)). Let Xn be the (normalized) number of individuals of some biological species present in year n. Then, the prescription (1.2.1) predicts the number of individuals in the following year n -I-1. The logistic map... 

A front corresponds to a traveling wave solution, which maintains its shape, travels with a constant velocity v, p x, t) = p(x - v t), and joins two steady states of the system. The latter are uniform stationary states, p(x, t) = p, where Ffp) = 0. For the logistic kinetics, the steady states are = 0 and jo2 = 1- While the logistic kinetics has only two steady states, three or more stationary states can exist for a broad class of systems in nonlinear chemistry and population dynamics with Alice effect, but a front can only connect two of them. To determine the propagation direction of the front, we need to evaluate the stability of the stationary states, see Sect. 1.2. The steady state jo is stable if P (fp) < 0 and unstable if F (jo) > 0. Let the initial particle density p x,0) be such that on a certain finite interval, p x,0) is different from 0 and 1, and to the left of this interval p(x,0) = 1, while to the right p x, 0) = 0. In this case, the initial condition is said to have compact support. Kolmogorov et al. [232] showed for Fisher s equation that due to the combined effects of diffusion and reaction, the region of density close to 1 expands to the... 

The resulting equation (15.18) is known as the logistic equation and has played a major role in efforts to understand population dynamics. It is analyzed in detail in many books on mathematical biology (e.g., Edelstein-Keshet, 1988). The key result is that for any initial population, the system approaches the single stable steady-state solution Pi = K. 

This difference equation is called a logistic map, and represents a simple deterministic system, where given a yi one can calculate the consequent point y% and so on. We are interested in solutions yi > 0 with 6 > 0. This model describes the dynamics of a single species population [32]. For this map, the fixed points y on the first iteration are solutions of... 

The model (1) has four parameters R, K, A, and B. As usual, there are various ways to nondimensionalize the system. For example, both A and K have the same dimension as A, and so either N/A or N/K could serve as a dimensionless population level. It often takes some trial and error to find the best choice. In this case, our heuristic will be to scale the equation so that all the dimensionless groups are pushed into the logistic part of the dynamics, with none in the predation part. This turns out to ease the graphical analysis of the fixed points. 

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