Nonlinear Birth-Death Models

Abstract

Consider now modeling the single population of sizeX(t)with population rates

$$ \lambda _X = \left\{ \begin{gathered} a_1 X - b_1 X^{s + 1} for X < (a_1 /b_1 )^{1/s} \hfill \\ 0 otherwise \hfill \\ \end{gathered} \right. $$

(6.1)

$$ \mu _X = a_2 X + b_2 X^{s + 1} $$

(6.2)

for integers >1. These are called nonlinear rates in ecological population modeling because the per capita rates, i.e.Ax/X, are obviously functions ofX, as discussed in Section 3.7. The a_iare called the intrinsic rates, and are interpreted as the per-capita birth and death rate coefficients for small initial population sizes, before population density pressures are of any practical consequence. Theb _i are the crowding coefficients which add density-dependency to the model. Their inclusion may in principle yield long-term, equilibrium solutions for the model.