 # Waves in Biological Populations

DOI: 10.4018/978-1-5225-9651-6.ch002

## Abstract

Self-regulating nonlinear waves in various biological populations are considered as moving attractors in excitable media. Mathematically, waves in populations are solutions of nonstationary parabolic systems of differential diffusion equations with source terms, and the velocity of the wave is an eigenvalue of the problem, and its profile is an eigenvalue function of the problem. There is no general exact method for solving such a problem. An approximate method for its solution is proposed (the semi-infinite reaction zone method), which essentially reduces to solving an algebraic system of equations. The method is used to calculate the waves in various biological populations. It is shown that there are two types of waves: a wave of conquest and a solitary wave. In all cases considered, formulas for calculating the velocity of the wave and its profile were obtained. One of the important examples considered is the analysis of solitary waves in populations of the herd locust.
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## Waves In The Logistic Biological Populations

To study the waves of biological populations, one will use the semi - infinite reaction zone method developed by the author in the study of the polymerization waves and combustion waves (Zhizhin, 1982, 1988, 1992, 1997 a, b, 2004a,b, 2008; Zhizhin, & Poritskaya, 1994; Zhizhin, & Larina, 1994).

The nonstationary equation for the change in the concentration N of individuals in the logistic population in a one - dimensional range, taking into account the chaotic mobility μ of individuals, has the form (Svirezhev, 1987)

(1)

Here F (N) is the function describing the local law of the population growth, and

(2) where B and D are the fertility and mortality functions.

Equation (2) describes the fact that the local law of the population growth is determined by two processes — birth and death. In the model of the logistic population it is assumed, that mortality D is a linearly increasing function of the concentration of individuals

Here is natural mortality. Increase in mortality with increasing concentration N is due to increased competition with limited resources (food, space, etc.). In addition, it is assumed that the fertility function B it is determined only by the physiological limits of fertility and is independent of N, i.e. B = m = constant, where m is the so - called natural fertility or fecundity. Given these assumptions and equation (2), equation (1) takes the form

(3)

Here is Malthusian parameter, and is the capacity of the medium, i.e. maximum possible concentration of individuals in the environment.

If the competition between individuals of the population is completely absent  , then the law of the local population growth takes a simple form . In this case, the population density increases indefinitely (the Malthus model of exponential growth) and the wave solution, as a transition from one equilibrium position to another, does not exist .

To find the stationary wave solution of equation (3), one can introduce the wave coordinate , here u is the wave velocity. One also introduce dimensionless variables Then equation (3) can be written as a system

(4)

## Key Terms in this Chapter

Special Point: The point at which the first derivatives of all phase variables with respect to the independent variable are equal to zero.

Solitary Wave: Wave motion that at each time instant is localized in a finite region of space and rapidly decreases with distance from this region.

Semi-Infinite Reaction Zone Method: A method that assumes a low reaction rate to the reaction front and a significant reaction rate after the reaction front to infinity.

Zero Isocline: The line at the points of which the first derivative of some phase variable with respect to the independent variable is zero.

Separatrix: A trajectory that separates qualitatively different types of trajectories from each other.

Gregarious Locust: The state of the locust in which the locust forms large flocks capable of long flights; the gregarious locust is characterized by bright coloration and aggressive behavior.

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