## Time Lag Models

Time-lag models attempt to create population growth forms which better reflect natural conditions because organisms rarely respond instantaneously to a change in the system. Introducing a time lag into a model can destabilize the model and cause dramatic fluctuations in population size, depending on the length of the time delay. The simplest time-lag model is a modification of the birth and death rate model (eqn [7]):

where b is the instantaneous birth rate, t is the current time, t represents the time delay (the N(t — t) term is the population size at time t — t), and d is the instantaneous death rate. This model assumes that d is related to the current population size, but that b depends on the population size at time t.

The Verhulst-Pearl logistic equation can be modified to incorporate time lags:

where t represents the current time and t represents the time delay that individuals experience before they are capable of contributing new individuals to the population. The dynamics of this model are controlled by values of the combination r t. The model will reach a stable equilibrium for 0< rr < e — 1 (e — 1 « 0.36788), damped oscillations for e — 1 < rr < ^/2, and a stable limit cycle for rr > ^/2.

Equation [13] can be further modified to incorporate more than one time lag:

where rg is a second time lag. The second time lag will further slow the rate of population increase, but will create patterns of growth similar to those in Figure 4, that is, the larger the time lag, the larger the amplitude of the oscillations.

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