ABSTRACT
The logistic model was first proposed by Belgian mathematician Pierre Verhulst (Verhulst (1838)):
dP
dt = aP
( 1− P
N
) , a,N > 0. (3.1)
Here a is the maximum growth rate per capita, and N is the carrying capacity. A more general logistic growth type can be characterized by a declining growth rate per capita function. However it has been increasingly recognized by population ecologists that the growth rate per capita may achieve its peak at a positive density, which is called an Allee effect (see Allee (1938), Dennis (1989) and Lewis and Kareiva (1993)). An Allee effect can be caused by shortage of mates (Hopf and Hopf (1985), Veit and Lewis (1996)), lack of effective pollination (Groom (1998)), predator saturation (de Roos et.al. (1998)), and cooperative behaviors (Wilson and Nisbet (1997)).
If the growth rate per capita is negative when the population is small, we call such a growth pattern a strong Allee effect (see Fig.3.1-c); if f(u) is smaller than the maximum but still positive for small u, we call it a weak Allee effect (see Fig.3.1-b). In Clark (1991), a strong Allee effect is called a critical depensation and a weak Allee effect is called a noncritical depensation. A population with a strong Allee effect is also called asocial by Philip (1957). Most people regard the strong Allee effect as the Allee effect, but population ecologists have started to realize that an Allee effect may be weak or strong (see Wang and Kot (2001), Wang, Kot and Neubert (2002)). Some possible growth rate per capita functions were also discussed in Conway (1983,1984). A prototypical model with Allee effect is
