Biological Waves: Single Species Models

Abstract

There is a vast number of phenomena in biology where a key element or precursor to a developmental process seems to be the appearance of a travelling wave of chemical concentration or mechanical deformation. Looking at almost any film of a developing embryo it is hard not to be struck by the number of wave-like events that appear after fertilisation. There are, for example, both chemical and mechanical waves which propagate on the surface of many vertebrate eggs. In the case of the egg of the fish Medaka a calcium (Ca⁺⁺) wave sweeps over the surface; it emanates from the point of sperm entry: we briefly discuss this problem in Section 11.6 below. Chemical concentration waves such as those found with the Belousov-Zhabotinskii reaction are visually dramatic examples (see Fig. 12.1 in the following chapter). From the analysis on insect dispersal in Section 9.3 in Chapter 9 we can also expect wave phenomena in that area, and in interacting population models where spatial effects are important. Another example, related to interacting populations, is the progressing wave of an epidemic, of which the rabies epizootic currently spreading across Europe is a dramatic and disturbing example: we study a model for this in some detail in Chapter 20. The movement of microorganisms moving into a food source, chemotactically directed, is another. The slime mold Dictyostelium discoideum is a particularly widely studied example of chemotaxis: we discuss this phenomenon later (see the photograph in Fig. 12.16 which shows associated waves).