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Journal of Theoretical Biology 241 (2006) 109119 An alternative formulation for a delayed logistic equation$

Summary: Journal of Theoretical Biology 241 (2006) 109­119
An alternative formulation for a delayed logistic equation$
Julien ArinoĆ,1
, Lin Wang2
, Gail S.K. Wolkowicz
Department of Mathematics & Statistics, McMaster University, Hamilton, Ont., Canada L8S 4K1
Received 10 September 2005; received in revised form 8 November 2005; accepted 9 November 2005
Available online 27 December 2005
We derive an alternative expression for a delayed logistic equation, assuming that the rate of change of the population depends on
three components: growth, death, and intraspecific competition, with the delay in the growth component. In our formulation, we
incorporate the delay in the growth term in a manner consistent with the rate of instantaneous decline in the population given by the
model. We provide a complete global analysis, showing that, unlike the dynamics of the classical logistic delay differential equation
(DDE) model, no sustained oscillations are possible. Just as for the classical logistic ordinary differential equation (ODE) growth model,
all solutions approach a globally asymptotically stable equilibrium. However, unlike both the logistic ODE and DDE growth models, the
value of this equilibrium depends on all of the parameters, including the delay, and there is a threshold that determines whether
the population survives or dies out. In particular, if the delay is too long, the population dies out. When the population survives, i.e., the
attracting equilibrium has a positive value, we explore how this value depends on the parameters. When this value is positive, solutions of
our DDE model seem to be well approximated by solutions of the logistic ODE growth model with this carrying capacity and an
appropriate choice for the intrinsic growth rate that is independent of the initial conditions.


Source: Arino, Julien - Department of Mathematics, University of Manitoba
Wolkowicz, Gail S. K. - Department of Mathematics and Statistics, McMaster University


Collections: Mathematics