Transient dynamics can be simulated by iterating the exponential step-ahead forecasting equation

When R > 0 (or births greater than deaths), the population grows at an increasing rate (exponentially):

When R < 0 (or births less than deaths), the population decays exponentially towards zero, or goes extinct:

When the logarithms of population density are plotted against time (graphs on the right-hand side), we obtain linear relationships with slope equal to the per-capita rate of change, R. Transient dynamics are said to be unstable because the population does not return towards its previous density but either grows towards some new density or declines towards extinction.

We can also have a stable solution under the highly unlikely condition that R = 0 (or births exactly equal deaths), then the population remains unchanged, or in equilibrium:

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