ABSTRACT

Consider the Volterra model for the population growth [99] of a species within a closed system governed by a nonlinear integro-differential equation

β du(t) dt

= u(t)− u2(t)− u(t) ∫ t 0

u(x)dx, (10.1)

subject to the initial condition

u(0) = α, (10.2)

where u(t) is the scaled population of identical individuals, t denotes the time, and β = c/(ab) is a nondimensional parameter in which a > 0 is the birth rate coefficient, b > 0 is the crowding coefficient, and c > 0 is the toxicity coefficient, respectively. For details the reader is referred to Scudo [99], Small [100], TeBeest [101], and Wazwaz [102].