Discretetime analogs of the exponential and logistic modelsExponential model analog:
where t is time measured in generations, and R is net reproduction rate. For monovoltine organisms (1 generation per year), R is the average number of offsprings per one parent. For example, in monovoltine insects with a 1:1 sex ratio, R = Fecundity/2. The dynamics of this model is similar to the continuoustime exponential model. Logistic model analog (Ricker):
The dynamics of this model is similar to the continuoustime logistic model if population growth rate is small (0 < ro < 0.5). However, if the population growth rate is high, then the model may exhibit more complex dynamics: damping oscillations, cycles, or chaos (see Lecture 9). An example of damping oscillations is shown below:
Complex dynamics results from a time delay in feedback mechanisms. There are no intermediate steps between time t and time t+1. Thus, overcompensation may occur if the population grows or declines too fast passing the equilibrium point. In the continuoustime logistic model, there is no delay because the rate of population growth is updated continuously. Thus, the population density cannot pass the equilibrium point.
